Machine Kinematics
solved, then drawn
Theme

The thread: Exact and approximate

Watt's straight line is not straight and Peaucellier's is. The difference is measurable, it took ninety years to close, and the error curve of an approximation is a more interesting object than the approximation.
What the machine draws, against where the polynomial vanishes. Two objects, found two ways. The thin line is the set where x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 is zero, walked over a grid with no mechanism involved. The marks are where the compiled machine's tracing point went, one per converged solve, over the 147 positions of its working arc. The machine's constraint set never mentions the polynomial, so evaluating it at each traced point is an independent check: the worst value over the whole arc is 3.7e-13. The arc is 1.30 radians of the driving angle and not the whole turn, and past that arc it draws something else. The curve as an equation

A demand that is an equation

Every field on this site is handed its demand geometrically — three positions, a sampled path, a ratio at each angle — and hands back a mechanism that is right at those places and approximately right between them. This one is handed a polynomial, and the mechanism that comes back satisfies it everywhere it moves.

 One routine, six strand systems. Every row is the same function: a list of bodies, each with a sense, handed to a routine that returns the tangent runs between them, the arcs on them, and the total. Nothing in it knows what a belt is, what a tackle is or what a tendon is. The right-hand column is what each row was checked against — a textbook formula, an integer, a convex hull's perimeter, a second route to the same length — and it is the reason the middle column can stay the same all the way down. positioned by solving, not by drawing. Members that pull

A member with no length of its own

Every link of a pin-jointed machine holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a rope and a tendon do neither: a strand's path is decided by the bodies it touches, and its constraint is an inequality that does nothing at all until it is taut.

 Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off. The problem backwards

The problem the other way round

Every essay before this one starts from link lengths and finds the motion. That is the reader's problem, because lengths are what a drawing shows. It is not the designer's problem, which is the reverse — and the reverse is hard enough that for a century the practical method was to look the answer up in a book.

 Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are. How many answers

Two circles, four answers

A four-bar with its crank held still is two circles, and two circles meet twice. Bézout's theorem says four. The two missing answers are not a rounding error and are not special to these link lengths — they are the same two points for every pair of circles ever drawn, and they are the beginning of a way of counting that has so far been done by hand.

 A simple planetary, as a graph. Members are vertices and meshes are edges. Each edge carries the two tooth counts and the body the two axes are stationary in — its carrier — and that third label is the whole of what makes an epicyclic different from an ordinary train. Write the mesh relation relative to the carrier and one formula covers both: an ordinary train is the case where every carrier is the frame. This train has 2 meshes across 5 members and 2 freedoms. More than one input

A ratio is a null space

Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.

 A flank nobody would draw, and the partner it forces. On the left, a driving flank made up on purpose: a rising curve with a nine-cycle wobble on it, chosen to be nothing in particular. On the right, in the driven wheel's frame, the shape that has to mate with it. There was a contact at all 121 sampled positions and the worst residual of the meshing equation was 3.13e-10. Conjugate action does not single out the involute — every profile has a partner that holds the ratio exactly, and the reasons for preferring one tooth form over another are all somewhere else. Generating back the other way returns the original flank to 2.33e-5 mm over 61 points. positioned by solving, not by drawing. The shape is the unknown

Any shape has a partner

Conjugate action does not pick out the involute. Hand the construction a flank invented on purpose to be nothing in particular and it returns a mate that holds the ratio exactly — so the question a tooth form answers is not whether it can transmit motion, and every real reason for choosing one is somewhere else.

 Two routes to the same number, along one sweep. The polynomial x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 evaluated at the tip of the arm as the second angle goes right round, by two routes that share nothing: once by putting the tip's coordinates into the polynomial, and once by adding up 5 cosines of whole-number combinations of the two angles. The two curves are drawn on top of each other and the worst gap between them is 8.9e-15. Where the line crosses zero is where the machine may stand: 2 crossings at this α, which is how many assemblies the compiled machine has at this driving angle. The curve as an equation

Every curve is a sum of cosines

Put the two angles of a two-link arm into any polynomial in x and y and what comes out is a constant plus a finite sum of cosines of whole-number combinations of them. Nine curves, three hundred random angle pairs each, and the two routes agree to 1.8 × 10⁻¹⁴.

 A point, its pole, and the centre it is turning about. The tracing point is on the coupler at (0.45, 0.5) of its length. The cross is the pole, the faint curve is the path the point traces over a whole turn, and the circle is the one that path is momentarily on — centre marked, radius 0.267. The point, the pole and the centre are collinear, which is not an accident of this position: a point's centre of curvature always lies on its own ray from the pole, and Euler and Savary's relation says where on it. Here that relation puts the centre 5.6e-16 of a unit from where differentiating the loop equation three times puts it. positioned by solving, not by drawing. The motion, not the mechanism

Every point has a centre

A point of a moving plane traces a curve, and near an instant that curve is a circle. Which circle is decided by one relation with two numbers in it — the same relation for every point of the plane at once, and it was written down in 1830 with no derivatives visible anywhere in it.

 Every path, in the plane of one unknown. The 16 tracked paths of the 3-RPR platform, projected onto the complex plane of x. Each curve starts at a solution of the start system and ends at a solution of the target or leaves the frame on its way to infinity. This run is γ random, start constants complex, and it found 6 solutions. How many answers

Following a root from a problem already solved

Homotopy continuation solves a system nobody can solve by deforming one that anybody can, and following every root as it moves. The whole method rests on the deformation being generic, and the folklore says that is what the γ-trick is for. Running all four combinations says the folklore names one of two places the randomness can live, and either will do.

 SCARA, as four numbers a joint. A Denavit–Hartenberg table describes each joint by the common perpendicular between its axis and the next one: how long it is, how much the axes twist across it, where along the first axis it meets, and at what angle. 3 of these 3 rows have no answer for the last two, because the axes they describe are parallel and two parallel lines have infinitely many common perpendiculars, all the same length, at every point along them. The arm is perfectly ordinary; it is the description that has run out. One path to the tool

Four numbers or a screw

An arm can be written down as four numbers a joint or as a line in space with a pitch on it. Both are minimal, both describe the same machine to the last bit, and one of them jumps by three hundred and fifty thousand when an axis is tilted by a millionth of a radian.

 Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance. What a joint is

Six, and no others

Eleven surfaces were handed to the same computation and six groups came out. A cone, a torus and an ellipsoid of revolution give the same joint; a scalene ellipsoid gives none; and the list does not grow when more surfaces are added, because a surface's symmetry group has to leave a two-dimensional set alone and only six groups can.

 The same rise, three ways. A 20-unit rise over 120°, then a dwell. Above, the displacement — all three do the same job and are hard to tell apart. Below, the acceleration, differentiated from those same curves. Constant acceleration peaks lowest at 5.556e-3 per degree² and arrives at the dwell with a step; cycloidal peaks highest at 8.727e-3 and arrives at zero. The smoothest law has the largest peak, which is the trade the whole of cam design turns on and which the displacement plot gives no hint of. Prescribed motion

The law that costs least is not the smoothest

Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.

 How straight, over how much of the stroke. The deviation from a straight line, as a fraction of the traced span, against how much of each mechanism's stroke is used. The vertical axis covers fifteen decades. Watt's and Chebyshev's linkages are excellent over a short stroke and degrade as more is used; Peaucellier's sits at the bottom of the plot at every fraction, because it is not an approximation. The gap at full stroke is about fourteen orders of magnitude, and it is the difference between a mechanism that is nearly right and one that is right. The paths points trace

The straight-line problem

Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.

 One moving pin, and the pivot it turns about. Choose any point of the moving body — this one at (-0.55, 0.5) in the body's own frame. In the three prescribed poses it lands in three places, and three points that are not in a line lie on exactly one circle. That circle's centre is where the fixed pivot has to be and its radius is how long the link has to be: here 1.0860, and all three images sit at that distance to within 10⁻¹². There is no iteration and no tolerance in the construction, because three points determine a circle exactly. The freedom is entirely in which point of the body to pick. The problem backwards

Three positions, and a circumcentre

The whole of three-position synthesis is one observation: a moving point occupies three places, three points that are not in a line lie on exactly one circle, and that circle's centre is where the fixed pivot has to be. No iteration, no tolerance, and every point of the coupler is a candidate.

 What one contact forbids, drawn as a place. A single contact on one edge of a square, and the whole plane coloured by what it permits. A rotation about a point is a twist, and a twist is permitted when it does not drive the part into the obstacle; because a rotation about (x, y) is affine in the point, the condition is a half-plane and the boundary is a straight line — the line through the contact along its own surface. On one side of it only anticlockwise rotations are permitted, on the other only clockwise, and the two together are the whole plane bar the line itself. So one contact rules out exactly half of what the part could do and leaves the other half untouched, which is why the count of contacts a hold needs is one more than the dimension rather than equal to it: the first 1 of them cannot leave nothing over. The picture is exact — the regions are clipped polygons, not a sampled grid. Contacts that only push

What one contact forbids

A rotation about a point is a twist, and a twist is affine in the point — so what a single contact permits is a half-plane of centres, with the boundary being the contact surface's own line. Reuleaux drew it in 1875 and it is exact rather than sampled, which is why every figure in this field is a picture of the plane rather than of a cone.

 Four tangents, and the two signs that choose between them. Two circles admit four common tangents, and a strand takes whichever one its two wrap senses name. Same sense at both ends — both centres on the same side of the strand — gives the two outer tangents, each 169.0444 mm long. Opposite senses give the two that cross between the circles, each 154.9193 mm. There is no search and no case analysis anywhere in this: the run's length is √(D² − Δ²) with Δ the signed radius difference, and changing one sense changes Δ from -18 to 70. positioned by solving, not by drawing. Members that pull

Where a strand leaves a body

A taut strand meets the surface it lies on at a right angle, and every book draws it that way. It is not a rule about strands: it is what being shortest looks like, and a family of paths that were never told about tangency has its minimum exactly there — 200.64346 mm against the construction's 200.64346.

 What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14. The curve as an equation

A circle costs one term

A line expands to two cosines, a circle to one, a general conic to six. A lemniscate is degree four and costs five; a general cubic is degree three and costs eight. What a curve costs is not its degree — it is how many frequency pairs its own symmetry fails to cancel.

 A dented shape, cut into 2 convex pieces. Ear clipping cuts the outline into triangles and Hertel–Mehlhorn then deletes every diagonal whose removal leaves both sides convex, which takes this crank to 2 pieces rather than the 4 the triangulation produced. The pieces tile the polygon to 1.8e-16 of its area, and — the part the areas cannot check — a point is inside the pieces exactly when it is inside the polygon, tested at 4,000 random points per shape with 0 disagreements. The gap between two parts is then the best answer over the pairs of pieces, which is why the decomposition has to be right rather than merely plausible. Links with a width

A shape with a dent in it

The separating-axis theorem is not approximately right about a non-convex shape; it is wrong, and it returns a confident number while being wrong. The repair is to cut the shape into convex pieces — and the shortcut everybody takes instead adds a hundred per cent more material.

 What a pattern that cannot fold leaves behind. The best a least-squares solve can do with the vertex closures, against the fold it is asked for. The Miura pattern closes at every angle, at the arithmetic's own floor — the line along the bottom is 10⁻¹⁵ and below. The same grid with its vertices moved by a tenth of a panel does not close at any angle at all: its residual starts at 5.9e-6 at the smallest fold and grows with it, and no seed and no number of iterations moves it. Both patterns have the same panels, the same creases, the same graph and the same developable vertices, and every one of those vertices folds perfectly well on its own. Many of one thing

Each one moves, and together they do not

Take the pattern a Miura sheet folds along and move every interior vertex by a tenth of a panel. Every vertex still folds on its own — each is a spherical four-bar with a freedom of its own — and the four of them together fold to no angle at all, with a residual that starts at six millionths and never falls.

 6 contacts, and the box lifts straight off. 3-2-1, six contacts. Each pad is a contact and each arrow the direction the box is free to move there — the inward normal, and the whole of what the contact contributes. The rank is 6, which is full: these 6 contacts are 6 independent constraints and a bilateral version of them would fix the box completely. The margin is nought, so the box is free to leave, and no amount of tightening the tolerances on where the pads are would change that. Rank and hold are different questions and this is the pair of pictures that separates them. positioned by solving, not by drawing. Contacts that only push

Four in the plane and seven in space

Six independent constraints fix a body in space and six contacts fix nothing, because d vectors can span d dimensions and can never positively span them. The minimum is one more than the dimension — and it is a floor rather than an answer: six contacts on a box held it in none of four thousand random arrangements and seven held it in twenty-one.

 Peaucellier's cell: exact straight-line motion from pin joints. The rhombus and the two long arms hold |OP| · |OQ| constant at 16 = 5² − 3², which is inversion in a circle about O. Inversion carries circles through the centre to straight lines, and the link CQ makes Q run on exactly such a circle — so P travels on a line, with no approximation anywhere. Measured over 160 solved positions the deviation is 6.5e-16 of the span, which is arithmetic noise rather than a small error. The paths points trace

Peaucellier and the exact answer

Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and on this site it comes out straight to 10⁻¹⁶ of its span.

 The Sarrus linkage at 0°. Two three-joint chains in perpendicular planes, joining a fixed plate to a moving one. The left chain's three axes are all parallel, so it allows the plate to move in its plane; the right chain's are parallel to a perpendicular direction and allow the plate to move in that one. What both permit is a straight line, and only a straight line. Six revolute joints in a single loop: Kutzbach says 0 degrees of freedom, the screw system has rank 5 and says 1, and the plate rises. Measured over 24 positions, its tilt never exceeds 2.5e-14 radians and it never leaves the axis by more than 6.8e-14 — exact, from pin joints, with no approximation anywhere in it. Out of the plane

Sarrus, and the straight line that is exact

The planar answer to the straight-line problem took two hundred years and arrived as an inversion cell with eight bars. There is a six-bar answer that is also exact, that was published eleven years before Peaucellier's, and that works for a reason with nothing to do with inversion — it leaves the plane.

 The lever of a simple planetary. Each member sits at a position on the lever fixed by the tooth counts alone, and its speed is the height of one straight line over that position. The line here is drawn through sun and ring; every other member is plotted where the train's null space puts it, and lands on the line exactly — the residual is zero in rationals, and 2.2e-16 once the coordinates have been rounded to doubles for the drawing. Where the line crosses the axis is the member that is standing still, and that is what a brake does: it pins the line to the axis at one position and leaves it free to pivot there. The planets are on the lever too, off the end of it, which is where they belong — they are members of the train and are not shafts anybody can reach. More than one input

The lever that is the gearset

The lever diagram of an epicyclic is usually offered as a mnemonic. It is exact, and the reason is a fact about the null space: a gearset whose frame carries no teeth can turn as a block, and that one motion supplies the coordinate every member is plotted at. Where the line crosses the axis is the member standing still, and the ordering of the members on the lever settles which gears are reductions and which run backwards, without a formula anywhere.

 What became of Bézout's paths. four-bar coupler pin: 2 of 4 paths arrived at a solution and 2 went to infinity; 3-RPR platform: 6 of 16 paths arrived at a solution and 10 went to infinity; Gough, generic: 80 of 1458 paths arrived at a solution and 1378 went to infinity. The surplus is not merely wasted — it is cheap: a path on its way to infinity is abandoned in a handful of steps, while every path that arrives is tracked in full. How many answers

The paths that leave

Bézout's number over-counts, and the over-count is enormous — 1,458 tracked paths for 80 solutions. The obvious response is to find a method that tracks only the paths that arrive. That method exists, it was built, and it is four times slower, because the surplus paths are not merely surplus. They are cheap.

 Lost motion on a 24-tooth ratchet. A pawl can only drop into a tooth, so an input that has moved by less than one tooth pitch has moved the output by nothing at all. On 24 teeth the pitch is 15.0°, and that is the worst case with one pawl. Two pawls offset by half a pitch halve it and three thirds it, because whichever pawl is over a root drops first. None of this is a manufacturing question: the numbers are the same on a perfectly made ratchet, which is what separates them from the lost motion an as-built measurement would. Motion that stops

The resolution is the pitch

A ratchet's step and a ratchet's error are the same number. Nothing about how well it is made improves that, more pawls divide it by a whole number, and the obvious remedy — cut more teeth — runs into a wall that is geometric rather than practical: at a tooth depth of 0.16 radii the construction stops at twenty-nine.

 What the wraps add up to, and what decides it. Four runs this field draws, with every wrap angle signed by the way the strand goes round its body. The right-hand column is their sum divided by a full turn, and it is a whole number every time — the turning number of a closed plane curve, arrived at by adding up a handful of angles that were computed one at a time from tangent lines. It is 1 for a loop that goes round its pulleys once and 0 for a crossed belt, whose two wraps are equal and opposite whatever the two radii are. The serpentine's idler contributes -25.3°, and the total is still exactly one turn: a tensioner lengthens the path without changing what the path is. Members that pull

The wraps add up to a turn

Every wrap angle in a closed run is computed on its own, from a pair of tangent lines that knows nothing about the others. Signed by which way the strand goes round, they add to exactly one turn — or to exactly nothing, for a crossed belt — and the integer is decided by the route rather than by any of the geometry.

 Undercutting, either side of 17.10 teeth. Five gears, drawn whole and scaled to a common pitch circle so the tooth counts can be compared by counting them. The dedendum sits a fixed 1.25 modules below the pitch circle and the base circle sits at r·cos α, so as the tooth count falls the base circle rises relative to the root. Below N = 2/sin²α = 17.097 it rises above it, and the part of the flank between them lies where no involute exists — the cutter removes it. The familiar rule says seventeen; the exact figure is 17.10, so seventeen undercuts slightly and eighteen is the smallest count that does not. The teeth are not to a common scale, because at a common module the small gears would be unreadable. Teeth

Undercutting, and the seventeen-tooth rule

Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way.

 Burmester's curves, contoured rather than drawn. With three poses, every point of the moving body works: three images, one circumcircle. With four, a point's four images are concyclic only if it lies on a particular cubic — the circle-point curve — and the fixed pivots those points want lie on a second cubic, the centre-point curve. Both are drawn here as contours of a measured quantity: at each point of a 150×150 grid, how far the fourth image misses the circle through the other three, contoured at zero. Points refined onto the contour are concyclic to 3.8e-15; points 0.47 away from it miss by at least 2.0e-1. The two curves are keyed in the legend and the four prescribed poses are drawn faintly for scale. Three poses leave a designer the whole plane; four leave a curve. The four prescribed poses are outlined faintly for scale, and both curves are clipped to the frame: the centre-point curve is a cubic with unbounded branches that reach 360 units on a mechanism three units across, and the part worth looking at is the part near the machine. The problem backwards

What the fourth position costs

With three prescribed positions every point of the coupler will do, and a designer is spoilt for choice. Add a fourth and the whole plane collapses to a curve — only points on a particular cubic have four images that lie on a circle, and the cubic is Burmester's.

 Three machines a protractor cannot tell apart. The same four-bar at 0.60×, 1.00×, 1.50×, drawn one inside another at the same crank angle. Every one of them puts its output link at 102.914064°, and the three readings differ by 2.8e-14° — which is the solver's floor rather than a difference. A protractor on the output link is reading a function of the ratios of the lengths, so it is the same function for every member of this family, at every crank angle, exactly. Whatever such an instrument recovers, it is not the size of the machine. Numbers that were measured

The direction no protractor can see

A four-bar's output angle depends only on the ratios of its lengths. That is a sentence anybody would agree to, and it has a consequence with a number attached: the vector of the four lengths is annihilated by every row of the machine's own identification Jacobian, to 7.6 × 10⁻¹⁵, at every pose, for ever.

 A tackle's ratio, differentiated rather than counted. Four parts of line between two blocks 260 mm apart, drawn as one strand over real sheaves. The ratio a tackle is sold with is the number of parts supporting the moving block, and it is the limit of the true velocity ratio rather than its value: the parts are not parallel, so each of them shortens by less than the lift. Differentiating the run's own length gives 3.9471 here, 1.32% short of 4, and the gap closes as the blocks separate. positioned by solving, not by drawing. Members that pull

A ratio that is a derivative of a length

A tackle is sold by counting the parts of line holding the moving block: four parts, four to one. Differentiate the strand's own length instead and a four-part tackle gives 3.927 with its blocks 220 mm apart and 3.617 at 90 mm — and the integer it is named for is a limit it reaches nowhere.

 Bennett's four-bar at 40°. Four bars, four revolute joints, and axes that are not parallel — a spatial four-bar, which Kutzbach counts at -2 degrees of freedom. Bennett's condition, sin α / a = sin β / b, makes the screw system rank 3 instead of 4, so the mechanism has 1. Driving the first joint through a full turn, 48 of 48 positions assemble. Orthographic projection, viewed from 40° azimuth and 24° elevation; dashed stubs mark the joint axes. Out of the plane

Bennett, and the condition that moves it

A spatial four-bar is immobile by every count there is, and generically it cannot even be assembled at more than isolated configurations. Bennett found the one relation between four lengths and two twists that makes it turn through a full revolution — and break the relation by two parts in a thousand and most of the travel is gone.

 the reflector, solved. A rhombus whose far vertex is held on a line through the pivot. One side is the input, the line is the mirror, and the other side comes out reflected in it — which is where negation, doubling and addition all come from. The relation it satisfies is (μ, θ) ↦ 2μ − θ, and across a sweep of 41 positions the worst departure from it is 1.3e-13 radians. Every joint here is the output of a Newton–Raphson solve on the bar lengths; nothing is placed by the formula the picture is about. The curve as an equation

Four bars that add two angles

A rhombus on two links from one pivot points along the bisector of their angle, exactly, because a rhombus has equal sides. Hold its far vertex on a line and it reflects instead. From those two facts come negation, doubling and addition — and every whole-number combination of two angles a compiled machine needs.

 What a Simpson gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 4 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such. More than one input

Holding a member chooses the ratio

A gearset offers a plane of motions and a shift element is one linear condition, so a gear is a line in that plane. Enumerate every brake and every clutch and a Ravigneaux's twenty-seven combinations collapse to seven ratios — eighteen of them being the same gear, because locking any two members at all locks the whole gearset solid. Neutral is not one of the seven, because neutral is not a gear.

 Different numbers of ternary links, and the same spectrum. Two of the 230 ten-link chains whose adjacency matrices have identical characteristic polynomials — identical in every one of the eleven coefficients — and which are not the same chain. They do not even share their assortment — 6×2 + 2×3 + 2×4 on the left and 4×2 + 6×3 on the right. Counting the ternary links tells them apart and the spectrum does not. That is worth pausing on: the spectrum is the more sophisticated invariant, it is the one that got written into the literature as a test, and here it is beaten by the first thing anybody would try. The polynomial both of them have is λ^10 − 13λ^8 + 52λ^6 − 4λ^5 − 76λ^4 + 8λ^3 + 32λ^2. The chain before the lengths

Right until the size nobody checked

The characteristic polynomial of a chain's adjacency matrix is a fingerprint that costs nothing and separates every six-link chain and every eight-link one. At ten links it fails on two pairs — and on one of them, counting the ternary links tells the two chains apart while the polynomial does not.

 The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell on this site. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake this site has already made once. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of. Motion that stops

The arc that is concentric with the pivot

A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell on this site is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.

 Six sizes, one freedom, and a count going the other way. Creases and constraints both grow as the square of the sheet's side, and they grow at different rates: two per panel against three per interior vertex. So the counted column runs (n − 1)(3 − n) and is positive at two, nought at three and increasingly negative after that, while the measured mobility is one on every row. The redundant column is the difference and it is exactly (n − 2)² — one at three, four at four, nine at five, thirty-six at eight. A twelve-by-twelve sheet of a hundred and forty-four panels is counted at minus ninety-nine and has a hundred repeated constraints, and it is the same mechanism as the smallest one on this table. Many of one thing

The freedom that survives repetition

A Miura sheet has one freedom at four panels and one at a hundred and forty-four, and the count runs the other way: plus one, then nought, then minus three, minus fifteen, minus ninety-nine. The gap between them is exactly (n − 2) squared, which is a hundred repeated constraints on a sheet with one degree of freedom.

 How far inside the hull the origin actually is. The same seven arrangements with their margins drawn rather than tabulated, because the shape of this chart is the argument: the quantity is not a probability and not a percentage, it is a distance — how far the origin sits from the nearest face of the hull of the contact rows, with every row a unit vector so the number is comparable across arrangements. The two that hold come in at 0.211 and 0.091; the five that do not come in at exactly nought, and they are drawn at nought rather than left off. A margin that falls smoothly to nothing is what makes this a measurement: an arrangement approaching one that lets go says so before it does. Contacts that only push

The test is a program, not a rank

Three independent routes to one yes-or-no: enumerate the escape cone's extreme rays by cross products, take the convex hull of the contact rows and ask where the origin is, or hand the whole thing to a simplex. They agree on every arrangement — and the first version of the third one reported a disc as held, which is the one part in the field that no number of contacts holds.

 The coordinates that do not move. Above: the four link lengths as the whole machine is scaled from 0.4× to 2.5×, four straight lines through the origin. Below: Freudenstein's K₁ = g/a, K₂ = g/c and K₃ = (g² + a² + c² − b²)/2ac over the same range, three horizontal lines whose total variation is 1.8e-15. Each of them is homogeneous of degree zero in the lengths, so the scale ray is a level set of all three at once — and the map from a four-bar's shape to its three K's is invertible, so they are not merely invariant but complete. The identifiable quotient of a four-bar's parameter space is three-dimensional, and this site has had its coordinates since its first essay on synthesis. Numbers that were measured

The coordinates the site already had

Three of a four-bar's four parameters are recoverable, so there are three recoverable quantities. They are Freudenstein's K's, which this site has used to design function generators for as long as it has synthesised anything — and the same 3 × 3 linear system, read backwards, identifies a machine from three measured angle pairs.

 The bound that says the sweep missed nothing. Every corner of every body is an affine function of its link's two joints, so its speed is the same combination of the joint velocities the mechanism already solves for — and the largest corner speed anywhere on this drive is V = 1.255 per radian. A distance between point sets is 1-Lipschitz in those points, so from each sample the gap can fall no faster than V: the fine lines are those cones. Where two cones cross is the least the gap can be between the samples, and at 60 samples that is 0.0383 — positive, so nothing was missed. The same bound refuses the twelve-sample sweep of the stud machine, where the bound is -0.281. Links with a width

A sweep that missed nothing

A swept clearance check looks at finitely many positions of a machine that has infinitely many, and cannot report what it did not look at. Here is a twelve-sample sweep declaring a machine clear by 0.007 while it is 0.010 inside a stud — and the bound that refuses to certify it.

 Colour by degree, recolour by neighbours' colours, stop when nothing changes. The cheap half of every isomorphism routine there is, and the half that does most of the work. Start by colouring each link with how many pins it carries. Then repeatedly recolour it with its own colour plus the multiset of its neighbours', until a pass changes nothing. On this chain the process ends with 3 classes of sizes 2, 2, 2, and two links of different colours are certainly different links — no relabelling can carry one to the other. What refinement cannot do is separate links that are alike to every local measurement, and that residue is what the backtracking search is for. It is also, exactly, why a spectral test fails: an eigenvalue is a global average over walks and has no more to say about two locally identical links than the refinement does. The chain before the lengths

Deciding that two chains are one

Two chains are the same chain when a relabelling of the links carries one to the other. Ten links admit 3,628,800 relabellings, and the census asks the question 26,335 times — so the answer is not a search but a rule that picks one labelling out of the graph itself, and asking whether the two strings match.

 Five positions, and what is left of the curve. Five prescribed poses of a moving body. With four of them, every point of the pale curve is a usable fixed pivot — a one-parameter family. The fifth pose is one more equation, and it leaves 4 points. Bézout's number for the system is 16; 4 paths arrive; 4 of those are real. Every pair of the 4 is a four-bar, so there are 6 candidate linkages and 2 of them reach all five poses in one piece and in order. 1 of the 4 pivots is too far away to draw in frame and is marked at the edge with its true distance — which is why some of the linkages have a bar twenty times the size of the body. The problem backwards

Five positions, and what is left

Three prescribed poses leave a whole plane of choices. Four leave a curve. Five leave four points, and finding them is the first thing in this site's synthesis field that a compass cannot do — it needs two cubics intersected, which is algebra rather than construction. Four points give six four-bars, and two of them can be built.

 What a Ravigneaux gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 5 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such. More than one input

Four speeds from two numbers

A Ravigneaux gearset has five tooth counts and gives seven exact ratios. Two of the counts do not appear in any of them — the short planet is an idler and its size is free — and the remaining three enter only through two dimensionless numbers, so the whole shift ladder of a four-speed automatic is a function of ring-over-sun and ring-over-the-other-sun. A Simpson three-speed is a function of one number.

 11 teeth, cut with three different shifts. A 11-tooth gear cannot be cut with a standard rack without the cutter eating into the flank near the root — the tooth is undercut, and what it loses is exactly the part that does the work. The fix is to hold the cutter further out by a fraction x of the module. Here the threshold is x = 1 − z sin²α / 2 = 0.3566, measured rather than quoted: at 0.347 the gear still undercuts and at 0.3566 it does not. What the shift costs is at the other end of the tooth. The tip thickness falls from 0.606 to 0.240 of a module, and a tooth shifted far enough comes to a point and breaks — so the technique has a ceiling as well as a floor. Teeth

Moving the cutter out

A gear with too few teeth is undercut by the tool that generates it, and the fix is to hold the tool further out. What that does to the tooth is easy to say. What it does to the pair is not what most readers expect — the two gears no longer mesh at the centre distance the sum of their radii would give, and the pressure angle they run at is no longer the one they were cut with.

 A quantity that is not there, refusing to converge. A second central difference of a function with a bounded second derivative settles as the step is halved; across a step in the first derivative it doubles, every time. The rising line is a mutilated gear at the instant its teeth engage, where the driven wheel goes from stopped to full pitch-line speed: its successive estimates grow by a factor of 2.000000, which is the signature and not an accident of the step. The flat line is a six-slot Geneva at the same point in its cycle, whose pin enters along the slot and whose acceleration is a number. This is the only way to report an acceleration that does not exist: not by quoting a large one, but by showing the measurement refuse. Motion that stops

The gear with its teeth cut away

Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.

 The measurement every gear cut since 1900 is a consequence of. Take a pair that is conjugate at its design centre distance, move the shafts apart, and ask what shape the driven wheel would have to be for the ratio to hold. The involute wants the same shape at every distance — 2.91e-8 mm at two millimetres out, which is the comparison's own noise floor — because its shape is fixed by its base circle and the centre distance is not one of that circle's arguments. The cycloidal pair is conjugate at nought and wants a shape 6.71e-3 mm different at five hundredths of a millimetre out, because its describing circle has to roll between two pitch circles that are no longer touching. A bearing that wears, a housing bored a little wide, a case that warms up: all of them are this axis. The shape is the unknown

The shape that does not mind where the shafts are

Two tooth forms, both exactly conjugate, both in use for centuries. Move the shafts five hundredths of a millimetre apart and one of them wants a different shape and the other does not — and that single measurement is close to the whole reason every gear cut since about 1900 is an involute.

 The twelve that were at infinity, coming back. Every solution of the platform's direct kinematics, plotted by how far from the origin it sits, as the six anchors are jittered. At no jitter the site's own platform has 56 solutions and the largest is at 25.8. At a jitter of 0.2 there are 80, and the extra ones arrive from far out — they were never missing, they were at infinity. How many answers

Twenty-eight, not forty

The general six-legged platform has forty poses for a given set of leg lengths, and this site has quoted that number beside a picture of a platform that has twenty-eight. Its anchors are arranged symmetrically, which makes it a special architecture, and the missing twelve poses are not missing. They are at infinity, and perturbing the anchors brings them back.

 Three linkages, one polynomial. Roberts's theorem says three different four-bars draw the same coupler curve. Here each one is traced, and each trace is fitted separately for the sextic that vanishes on it — on a common normalisation, or the comparison would be between three polynomials in three coordinate systems. The twenty-eight coefficients agree across all three to 5.0e-7. The three traces are drawn on top of one another and the curve is the same object each time; the test shares nothing with the construction that produced the cognates, which is why it is a test. The paths points trace

Three linkages, one equation

Roberts's theorem says three different four-bars draw the same coupler curve. Fitted separately for the sextic that vanishes on each trace, on a common normalisation, the twenty-eight coefficients agree across all three to 5 × 10⁻⁷ — a test of the theorem that shares nothing with the construction the cognates came from.

 A cam is the envelope of a roller. The follower's roller drawn in the cam's frame at 15 angles of the cam: the roller slides in a straight guide and the cam turns under it, so from the cam's point of view the roller travels round it on a path set by the lift law. The cam's surface is the envelope of those circles — the same computation as a rack generating a tooth, with a circle instead of a straight edge and a lift law instead of a rolling condition. The cams field builds the same surface by offsetting the roller-centre path inward by the roller radius, and the two agree to 4.1e-4 mm. positioned by solving, not by drawing. Prescribed motion

A cam is a conjugate pair

A cam is built by offsetting the path of the follower's centre inward by the roller radius. It can also be built by asking what shape stays in contact with a circle that slides in a stated way — the same computation that cuts a gear tooth — and the two surfaces agree to sixteen millionths of a millimetre.

 Backlash is what the centre distance buys. A 20-and-40-tooth pair, module 1, run at centre distances either side of the one at which the two teeth exactly fill the circular pitch. Backlash is measured from the drawn tooth thicknesses — no involute equation appears in the calculation — and plotted against the textbook linearisation j = 2 Δa tan α_w. Left of zero the teeth interfere and the pair cannot be assembled at all. So backlash is not slop and it is not wear: it is a quantity a designer buys with a centre distance, and buying none of it means specifying a centre distance that has to be exact at every temperature. Teeth

Backlash is an allowance

A gear pair with no backlash cannot be run, and a pair with the wrong amount cannot be assembled. It is bought with a centre distance — 0.03 too far apart on a 30 mm centre buys 0.022 of it, which is a quarter of a degree at the pinion — and the textbook formula that says so is right about the slope and drifts 4.5% at a centre-distance error nobody would accept anyway.

 What it costs to multiply an angle. Multiplying an angle by n is done by doubling and adding — 2k is one reflector applied to k, and 2k+1 is one addition of k's result and the original — so the cost follows the binary expansion of n and not n itself. Eight costs three doublings; seven costs six gadgets, twice as much, for a smaller number. The other line is what a chain of n−1 additions would cost. The difference matters because the highest multiple a curve of degree d needs is d, so the arithmetic in a compiled machine grows like d log d while the carrying grows like d⁴ — which is why the arithmetic is not what makes these machines large. The curve as an equation

Doubling is cheaper than adding

Multiplying an angle by eight costs three gadgets and multiplying it by seven costs six. The cost of an integer multiple follows the binary expansion of the integer and not its size — which is why the arithmetic in a compiled machine grows like d log d while everything else grows like the fourth power.

 Six gearsets, three conditions. Every one of these can be drawn, and five of the six are drawn in some textbook or other. The columns are the three conditions a planetary has to satisfy: that a whole planet fits between the sun and the ring, that the sun and ring teeth add to a multiple of the planet count, and that the planets clear each other. The last column is how far out of mesh the worst planet station is, in teeth — a quantity that is zero or is not, and that no drawing shows, because a drawing of a planetary at this scale draws circles. Drawn wrongly

The gearset that could not be assembled

A planetary drawing shows a sun, a ring and three or four planets between them, and if the circles are the right sizes at the right stations it looks right. The condition that decides whether the second planet can actually be dropped in is arithmetic — the sun and ring teeth must add to a multiple of the planet count — and it appears in no drawing, at no scale, in any style.

 A lower bound that happened to be tight. The site's own Gough platform at its home pose. The search starts Newton from a spread of guesses and reports what it lands on: 16 from 400, 16 from 1200, 16 from 4000. Tracking every one of the 1458 Bézout paths says there are 28 poses in the complex numbers and 16 of them are real. The search was right. Nothing available to the search could have said so. How many answers

The search that was right

This site has reported sixteen assemblies for its Gough platform and labelled the number a lower bound found by search, everywhere it appears. Tracking every path says there are twenty-eight poses and sixteen of them are real. The lower bound was tight. Nothing available to the search could have said so, and a second method that was supposed to settle it turns out to have the same defect one level up.

 The steps, against the ones the rule asks for. The design rule everybody quotes is that the steps between gears should be equal in ratio, so that the engine returns to the same speed after every shift. That makes the sequence geometric, and the ideal step for this spread over this many gears is 1.5324, marked. The steps a gearset actually gives are not free: the whole sequence is a function of the tooth counts, so once the top and bottom are chosen there is nothing left to spend on the middle. The worst step here is off the ideal by 10.8%. More than one input

The steps are not free

A gearbox is supposed to have equal steps between its gears, so that the engine returns to the same speed after every shift. A gearset has one or two numbers to spend on three or four gears, so from the third one the steps are a consequence rather than a choice — and asking for them to be equal turns out to be a quadratic whose root is the golden ratio, realised in tooth counts by consecutive Fibonacci numbers.

 A four-bar that computes log₁₀. The output rocker's angle, mapped back into y, against the input crank's angle mapped back into x. The pale curve is log₁₀ x and the solid one is what the linkage does. They agree exactly at the 3 precision points and nowhere else; the largest disagreement over the range is 0.2232° of rocker, which is 7.47e-4 in y. The chebyshev spacing put the precision points at 1.0670, 1.5000, 1.9330. The problem backwards

Three problems called synthesis

Prescribing a path, prescribing a whole pose, and prescribing a relation between two angles are three different problems with three different counts, and the word synthesis covers all of them. The third has a property the others do not — eliminate the coupler angle and the design equation becomes linear, so a four-bar that computes a logarithm falls out of a 3 × 3 solve with no iteration at all.

 What another measurement is worth. The smallest singular value of the identification Jacobian — the observability of the worst-recovered parameter — against how many poses were measured. The lower curve takes poses evenly round the crank; the upper one chooses each next pose to make this number as large as it can. Both rise steeply and then flatten: the 13 evenly-spaced poses already give four fifths of what 20 give. Chosen poses reach at 4 what even spacing needs 5 to reach. The flattening is not diminishing returns on accuracy — it is the information in a pose being a direction in a three-dimensional space, which a handful of well-spread poses already spans. Everything after that is averaging noise, which is a different gain and improves as the square root. Numbers that were measured

How many poses are enough

Three readings determine a four-bar's shape and the thousandth adds almost nothing. The rank is reached at three because there are three parameters, and everything after that is conditioning — which is a different quantity, improves for a different reason, and stops improving much sooner than anybody expects.

 Free to spin, and it cannot turn at all. An ellipse of semi-axes 1.4 and 0.9 inside four flat walls that touch it at the ends of its own axes. Every normal points at the centre, so every row's moment is nought and the four rows span two dimensions rather than three: the cone of permitted twists is the whole spin axis, a line through the origin, and the first-order answer is that the part is free to turn either way. Drag the angle and watch what happens. The ellipse's reach in the direction of the top and bottom walls is √(a²sin²θ + b²cos²θ), which is smallest at θ = 0 and grows from there, so any rotation whatever drives it into both of them — by 0.016 at this angle. A nullity is a candidate and not a motion, and this is the shape of case the fields before this one could not produce: not a mechanism at a singularity, but an ordinary part in an ordinary pocket. positioned by solving, not by drawing. Contacts that only push

Free to turn and unable to

An ellipse in a pocket the size of its own bounding box has four contacts whose rows span two dimensions, so the cone of permitted twists is a whole line and the first-order answer is that it spins both ways. It cannot turn by any amount whatever: the penetration grows as the square of the angle, with a fitted exponent of 1.9944, and a circle in the same pocket turns for ever.

 Four of the sixteen cannot be positioned without a solver. For every chain, every way of choosing a frame and a driven link pinned to it, and for each the decomposition into Assur groups. All dyads counts the choices whose groups are all two links — those are the mechanisms a draughtsman can position with a compass, two circles at a time. The last column is the one that matters: chains for which no choice of frame and input is all dyads, so every way of driving them leaves a group of four or more links that has to be solved as a single system. At eight links there are 4 of them and at ten there are 90. This site has run a Newton solve on every mechanism it has ever drawn, and it has always been possible to read that as convenience. On these it is not. The chain before the lengths

Eight ways to drive it, and one machine

Bolting a link down is half the decision; the other half is which link carries the input. A four-bar has eight frame-and-input pairs and exactly one of them is a distinct machine — and across the eight-link census 320 listed pairs collapse to 153.

 One member of the family is a pair of wheels. The same demand — one plus 0.4 sin φ — scaled by a constant, and for each the angle the driven wheel is out by after one turn of the input. It has to be zero, or the teeth do not line up with themselves and there is no wheel. Only the unscaled member closes; five per cent either way leaves the output 18° out, which is a third of a tooth on a thirty-tooth wheel and a mechanism that seizes on its second turn. Non-circular gearing is a search rather than a drawing for exactly this reason: the closure condition is one equation on a whole function, and almost no function satisfies it. The shape is the unknown

The demand that cannot be met

Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.

 The shape is what the demanded arm implies. A strand leaving a convex body runs along a tangent, so the only thing about the body the strand can feel is the perpendicular distance from the axis to that tangent — the shape's support function h(ψ). Ask for a rate and you have asked for h, because turning the body by dψ pays out h dψ; the shape then comes back from h with no solve at all, as h(ψ)û + h′(ψ)û⊥. This one was asked for h₀(1 + 0.3 cos 2ψ) with h₀ = 34 mm. The strand drawn here leaves the axis at a perpendicular distance of 24.1286 mm, which is what the demand asks for at this angle. positioned by solving, not by drawing. Members that pull

The drum that is not round

The only thing about a body a strand can feel is the perpendicular distance from the axis to the tangent it leaves along. Ask for a rate and you have asked for that distance at every angle — and the shape comes back from it with no solve at all, unless the demand exceeds 1/(n²−1), at which point there is no shape.

 How wrong the trapezoid is, arm angle 15.6°. The difference between the outer wheel's angle and the angle that would put all four wheels on one circle. It is zero at straight ahead by construction — both wheels point forwards — and it reaches 2.06° at 35° of lock. "One hundred per cent Ackermann" names a condition this linkage meets at 1 angle and nowhere else, and no four-bar can do better than a handful: the condition is not a rational function of the crank angle, and the linkage is. Machines you have met

The steering that is never right

For four wheels to roll without scrubbing, the two front wheels must point at different angles, and the relation between them is a cotangent condition no four-bar can satisfy. The trapezoid under every car meets it at straight ahead and, if the arm angle is chosen well, at exactly one other angle — 0.34° out at worst instead of 2.06°.

 Ball's point at 66°. Two curves and one point. The circle is the locus of points whose path is momentarily straight; the cubic is the locus of points whose path curvature is momentarily not changing. A point on both has a path that is straight and staying straight — four-point contact with its own tangent line, one order better than anything else in the plane. That is Ball's point. Walking the circle and looking for a zero of κ′ turned up 1 sign changes here and 1 of them survived being checked for continuity — the others are the asymptote the pole itself produces. Its measured order of contact with the line is 3.95. positioned by solving, not by drawing. The motion, not the mechanism

The straightest point there is

Where the circle of points going straight meets the cubic of points whose curvature is standing still, there is one point whose path is straight and staying straight. Neither Watt nor Chebyshev put their tracing point there, and moving Watt's to it more than halves the error over the whole stroke.

 How far back the wheel is pushed. Recoil against the part of the pendulum's swing that happens after the tooth has landed. The arc cut concentric with the pallet arbor is the flat line on zero, and it is zero as a matter of arithmetic rather than of smallness: rotation about the arbor carries that arc into itself, so the tooth's resting place does not move and every sampled value is the same double. Every other face rises. The flat face with no draw at all is the interesting one — it starts at zero and curves, because it agrees with the arc to first order and not to second, which is precisely the amplitude sensitivity a deadbeat exists to remove. Motion that stops

The wheel that goes backwards

While a pendulum finishes its swing the escape wheel is doing something, and what it does is decided entirely by the shape of the face the tooth is resting on. An arc about the pallet arbor sends it nowhere — not nearly nowhere, the same double at every sample. A flat cut tangent to that arc is dead at exactly one point of itself, and it is the one point the tooth never rests on.

 The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; uniform: worst 0.3415°, RMS 0.2296°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for. The problem backwards

Where the precision points go

A linkage that matches a function at three points is wrong between them, and where the three points are put decides how wrong. Chebyshev spacing cuts the worst error by a third against even spacing, for free — and the reason has nothing to do with mechanisms. It is a fact about a polynomial the linkage has never heard of.

 Six words, and the shortest of them. Every path a car that may not reverse and may not turn tighter than R can take between two placements is one of six shapes: three arcs, or two arcs with a straight between. All six are drawn; the shortest is LSR at 2.2557 R and the longest is RSR at 14.788 R. A numerical shooting solve that shares no line of code with the closed forms returns 2.2557 R, which agrees to 4e-16. Wheels, and where they may not go

A circle for the first millimetre

The shortest path for a car that may not reverse, from here to a point one millimetre to the side at the same heading, is 31.417 m for a five-metre turning radius. The shortest path to a point twenty metres to the side is 31.416 m. The cost of going sideways is not monotonic in how far sideways, and below a crossover at 2.956 R it is exactly 2πR + δ.

 A dwell is a measurement, not a stop. The Stephenson six-bar's output against a full turn of the crank, with the four-bar it is built on for comparison. Inside a band of ±1° the six-bar's output holds still for 145.8° of crank and the four-bar's for 41.9°. The dwell comes from a stretch of the coupler curve that fits a circle of radius 3.006 to within 2.96e-3, and the arm from the coupler point is that radius. Nothing here stops; it moves less than the band. Linkages

A dwell made from a curve

Parts of a coupler curve are very nearly circular arcs. Put a link of the arc's own radius on the coupler point and its far end stands almost still while the point runs along it, so the output dwells — 146° of crank inside a one-degree band, against 42° for the four-bar it is built on. A dwell linkage does not stop. It moves less than the tolerance, and how much less is a number.

 A ratio with no steps in it. Two pulleys whose sheaves slide on their shafts, and one belt. Pushing the primary's sheaves together makes the belt ride further out; the secondary's radius is then not a choice, because the belt is a fixed length and its length over two pulleys at a fixed centre distance is a function of both radii. So the secondary's radius here is the root of that equation, solved rather than assumed, and the drawn belt is the length it is supposed to be to 0.0e+0 mm. Ratio 1.000, with the two radii adding to 110.00 — a number the received rule of thumb says should not change and which changes by 2.6% across the travel. More than one input

A ratio with no steps in it

Push a variable pulley's sheaves together and the belt rides further out. The other pulley's radius is then not a choice — the belt has a fixed length — so it is the root of an equation, solved rather than set. The rule of thumb that says the two radii add to a constant is true to first order and wrong by 7.4% of the ratio at full shift, and the departure has a closed form.

 How far a joint-space move bows off the line. Every joint runs from its start value to its end value at a constant rate — the simplest possible move, and the one that can never be refused, because every point along it is a set of joint values and every set of joint values is a pose. The tool does not travel in a straight line while it happens. It bows away by 402 mm over a move of 1223 mm, which is 32.9% of the distance travelled and enough to hit something that the straight line would have missed. One path to the tool

A straight line at constant speed

Run every joint from its start value to its end value and the tool bows 402 mm off the line between them. Insist on the line instead and the arm will follow it — until the path passes near a singularity, where the joint rates a metre a second demands grow as one over the distance, measured at an exponent of −1.010.

 Every curve in the catalogue, compiled and counted. The bar count is not a formula: it is the number of bar constraints the compiled mechanism actually carries, asked of the object rather than predicted. A line costs five bars and a quintic four hundred and thirteen, which is the field's central number — exactness is paid for in size. The all-revolute column replaces the one prismatic pair that closes the chain with a Peaucellier cell, which costs seven bars whatever the curve, so it is more than half the machine on a line and a rounding error on a quintic. The last column is the polynomial evaluated at the traced point, worst over each machine's working arc. The curve as an equation

Five bars for a line, four hundred for a quintic

Nine curves compiled and counted: a line at five bars, a circle at eleven, a lemniscate at fifty, a general quintic at four hundred and thirteen. The growth is a fourth power of the degree, and three quarters of the largest machine is not computing anything at all.

 Four positions brought together. Burmester's construction for four prescribed positions gives a curve of points that can be fixed pivots. Bring the four positions together and it has to become the cubic of stationary curvature, because a circle through four coalescing positions is a circle of four-point contact. Measured along one ray from the pole: the finite curve crosses at one radius and the cubic at another, and the gap between them falls as the square of the spread — fitted exponent 2.024, and 7.9e-5 coupler lengths at a spread of 0.01 radians. The finite construction is the same concyclic test the site's four-position synthesis is built on, so this compares that machinery against the infinitesimal one rather than against a re-implementation of either. The motion, not the mechanism

Four positions brought together

Burmester's construction finds the points of a moving plane whose four prescribed positions lie on a circle. Bring the four positions together and the curve it draws has to become the cubic of stationary curvature — and the gap between the two closes as the square of the spread, with the exponent measured rather than assumed.

 Free in every direction, and it cannot get out. A disc of radius 1 among 3 point obstacles on a circle of radius 1.100. The shaded discs are the obstacles grown by the part's own radius, which is what the part's centre may not enter — the configuration space, and for a round part it is the plane itself. The part is caged when those grown discs overlap enough to close a ring around it, which happens below R = 1/sin(π/3) = 1.154701, and here it does. At every configuration inside the cage the part is free. The three normals all point at its centre, the rank of its rows is two, the escape cone is a whole line, and none of that has anything to do with whether it can leave. A hold is a statement about velocities at one configuration; a cage is a statement about where a finite motion can go, and the second does not follow from the first in either direction. positioned by solving, not by drawing. Contacts that only push

Free at every instant and going nowhere

Three points on a circle of 1.1 radii around a unit disc leave it free in every direction at every configuration — rank two, margin nought, the whole plane of centres shaded — and it cannot get out. The threshold is 1/sin(π/n), which is 1.154701 for three, and a flood fill of the free space agrees with the formula at every radius sampled.

 The network where counting works. 5 scissor units, each two bars pinned at their middles, with consecutive units sharing their end pins. Sixteen bodies and twenty-two pins at eight units, and Grübler's 3(n − 1) − 2j gives 4 — three rigid motions and one internal freedom — against a measured nullity of 4 and no redundant constraints at all. That holds at one unit and at sixteen. The tong is here to make the field's point in the direction nobody expects it: a network is not a place where counting fails, it is a place where counting stops being checkable by hand, and one of the two assemblies that spans this plate is counted perfectly. Span 8.1388 mm at this opening, which is 5 times one unit's 1.6278. positioned by solving, not by drawing. Many of one thing

One input at one end

A lazy tong's reach is 2nL cos θ — exactly, to a part in a million million, over eight sizes and sixty openings — so it multiplies its input by the number of units. It multiplies everything else by the same number, including the part of the drawing nobody wanted multiplied: a unit cut a hundredth of a radian out puts a tong of thirty-two units 0.374 out at the far end.

 One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything. Motion that stops

The angle that holds the lock

A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself: the tooth's push aims straight at the pivot and its moment is zero, so the smallest disturbance opens the lock. Tilt the face and the moment becomes ρ sin δ — and the same tilt, by virtual work, is exactly the rate at which the wheel is driven backwards. Draw and recoil are one angle read twice.

 The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°; minimax: worst 0.2105°, RMS 0.1463°. The error is zero at each precision point by construction and nowhere else. minimax has the smallest maximum here, and which curve is best depends entirely on which measure is asked for. The problem backwards

The linkage that is only nearly right

Stop demanding that a linkage pass exactly through three points, and ask instead that it be close everywhere. Three linkages result, none of them passing exactly through anything, and each is the best by a different measure — the least-squares fit beats the interpolant on average error and loses to it on the worst case. An optimiser gives exactly what it was asked for and nothing else.

 The radius a winch works at is not a property of the winch. A drum of core radius 30 mm filling with line 6 mm thick, one layer at a time. Line speed per turn is 2πr, and r is the radius of the layer being wound rather than anything about the machine: it runs from 207.35 mm on the first layer to 395.84 mm on the 6th, a factor of 1.909. A winch quoted at one speed is quoted at one layer, and which one is rarely said. positioned by solving, not by drawing. Members that pull

The radius a winch works at

Line speed per turn is 2πr, and r is not a property of the winch. It is the radius of whichever layer is being wound, so a six-layer drum runs from 207.35 mm per turn to 395.84 — a factor of 1.909 with nothing about the machine changed, and a length that is quadratic in the turns rather than proportional to them.

 A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding. Prescribed motion

A lift is a size and a law is a shape

A cam's motion law is a dimensionless function of a dimensionless argument, so everything the cam field says about laws — which has the least acceleration, which has an impulsive jerk, which one costs least — transfers between cams of any size. What does not transfer is anything with the lift or the base circle in it.

 What another measurement is worth. The smallest singular value of the identification Jacobian — the observability of the worst-recovered parameter — against how many poses were measured. The lower curve takes poses evenly round the crank; the upper one chooses each next pose to make this number as large as it can. Both rise steeply and then flatten: the 13 evenly-spaced poses already give four fifths of what 20 give. Chosen poses reach at 4 what even spacing needs 5 to reach. The flattening is not diminishing returns on accuracy — it is the information in a pose being a direction in a three-dimensional space, which a handful of well-spread poses already spans. Everything after that is averaging noise, which is a different gain and improves as the square root. Numbers that were measured

What another measurement is worth

The observability of a four-bar's worst-recovered parameter goes 0.118, 0.162, 0.188, 0.208 — and then keeps going up by less and less until adding a pose changes the fourth decimal place. The flattening is not diminishing returns on accuracy. It is a space of fixed dimension being filled.

 A parallelogram's sextic is a circle and a quartic. The parallelogram with ground 4, crank 1.5, coupler 4, rocker 1.5, coupler point at u = 0.45, v = 0.50. Its eliminated sextic divided by the circle of radius 1.5000 about (1.800, 2.000) leaves a remainder of 6.3 × 10⁻¹⁶ of its largest coefficient, and the quotient is a quartic whose leading form is exactly a multiple of (x² + y²)². Of the configurations solved across a full turn on both assemblies, 720 are on the circle — the machine as a parallelogram, coupler parallel to the ground — and 720 on the quartic, the machine crossed. The two factors meet at four finite points: 2 are change points, marked solid, where one configuration belongs to both; the other 2 are places the two drawings merely cross. The paths points trace

A sextic that comes apart

A parallelogram chain's coupler curve is not one curve. Its sextic divides exactly by a circle, leaving a quartic, and each factor is one of the two things the machine can do. The division leaves rounding and nothing else, a coupler one millionth too long leaves a remainder a million times larger than rounding, and the two factors meet at the two places where the machine has to choose.

 What degree a coupler curve actually is. A four-bar's coupler point traced at 602 solved positions, then asked which polynomials vanish on those points. For each degree the design matrix's smallest singular direction is taken and its worst residual plotted. Degrees four and five leave nothing near zero; degree six drops by ten decades and degree seven buys nothing further. The textbook sentence — a coupler curve is a sextic — comes back as a measurement, and the decision is made by the gap between the smallest singular value and the next, which here is a factor of 2.8e+3 — at 15, 21, 28 and 36 monomials respectively. How many answers

The equation a four-bar satisfies

Every textbook says a coupler curve is a sextic. Traced at six hundred solved positions and fitted at degrees four through seven, the answer comes back as a measurement: nothing vanishes below six, degree six drops by ten decades, and degree seven buys nothing — with the decision made by a gap of 2.8 × 10³ rather than by a residual.

 A chain is a strand with a smallest piece. 53 teeth driving 11 at a pitch of 12.7 mm. The taut strand is drawn as this field computes it; inside each pitch circle is the polygon the pins actually sit on. The two agree on the straight spans and not on the wraps, and the gap is what the applied field's chordal action is: the effective radius swings between R cos(π/n) and R within every tooth, which is 4.05% on the small sprocket and 0.176% on the large one. A strand has no such number, because a strand has no pitch. positioned by solving, not by drawing. Members that pull

A chain is not a strand

A chain has a smallest piece, and two things follow that no continuous model can have. Its pins sit on a polygon, so the radius that matters swings by 4.05% within every tooth of an eleven-tooth sprocket — and its loop must contain a whole number of pitches, so the centre distance that closes it comes in steps of 6.4834 mm.

 404 ways to gear 60 to 1. Each dot is a pair of wheels and a pair of pinions whose ratios multiply to exactly 60, with wheels of 40 to 120 teeth and pinions of 6 to 16 leaves. The line is the hyperbola every exact answer must lie on; the dots are the ones that lie on it at whole numbers of teeth, and there are 404 of them. In a single pair there are 0 — a ratio of sixty needs a wheel of 360 teeth against a pinion of six, and nobody cuts that. Exactness is decided in integers here and never by comparing floating-point ratios, which is the one way a search like this quietly returns answers that are merely close. Teeth

A clock is a factorisation

A going train's job is a ratio and its parts are tooth counts, so whether a clock can be built is whether a number factorises inside the counts a wheel-cutting engine will cut. Sixty has 404 answers in two pairs and none at all in one. The ratio between a sidereal day and a mean one has none in any number of pairs, and the best two-pair train is out by a sixth of a second a day.

 One planet shaft, two centre distances. A compound epicyclic gets its enormous reduction from two meshes whose tooth counts are nearly in the same proportion. The two planet gears are on one shaft, so their axes are at one radius — and at a common module the two rings ask for radii that differ by 0.50 of a tooth. The exactness the reduction is famous for is bought with a pair of meshes running away from the centre distance they were cut at, and the difference is made up by profile shift — the same correction the teeth field applies for a different reason. It is not a rounding: it is the mechanism's own condition, and it is the reason a catalogue reduction of this kind comes in a short list of tooth counts rather than in any combination. More than one input

A hundred to one from a difference of one

A harmonic drive reduces by a hundred to one in a single stage with two gears in it, and the hundred is the flexspline's tooth count divided by the two teeth the circular spline has more than it. The same null space that answers a planetary answers it. What each of the three single-stage reductions pays for that arithmetic is different, and the compound epicyclic's price is a pair of meshes whose centre distances differ by half a tooth.

 The cylindroid at 75° and 1.00 apart. Every screw in the two-system spanned by two revolute axes 1.00 apart along their common perpendicular and 75° out of parallel, drawn as its own axis. The axes sweep a ruled surface — the cylindroid — and each generator carries a pitch, running from -0.384 to 0.652 and reaching its two extremes on the two principal screws, which cross at a right angle at the centre. The surface is 1.035 long along its own axis, which is exactly the spread of the pitches: a cylindroid is as long as its pitches are far apart. Out of the plane

The smallest screw system has a shape

Add two screws together in every proportion. The results do not scatter — their axes sweep a ruled surface, with the pitch varying along it between two extremes reached at right angles to each other. It is a picture nobody would guess from the algebra, and it is the object that says what two joints between two bodies leave free.

 The four eight-link chains a compass cannot position. Every one of the twenty ways of choosing a frame and a driven link, on each of these four chains, leaves a group of four links or more that has to be solved as one system. There is no order in which they come apart two at a time, so there is no ruler-and-compass construction for any of them and no closed form for their positions. They are numbers 1, 3, 4, 10 of the sixteen, and they do not share an assortment: 4×2 + 4×3 and 5×2 + 2×3 + 1×4 both appear. Three of the four are among the most symmetric chains in the census — automorphism groups of 16, 8, 8 against a median of three across the sixteen — which is the direction one would guess, since a symmetric chain has few genuinely different places to attach a driven link. The fourth has an automorphism group of 2, so symmetry is a tendency here and not the reason. The chain before the lengths

Four that a compass cannot reach

Twelve of the sixteen eight-link chains can be positioned two links at a time, from at least one choice of frame and input. Four cannot be positioned that way from any of their twenty choices — and at ten links ninety of the two hundred and thirty are in the same position.

 The smallest mechanism that is not one. Two bars from a free joint to two pinned ones, with the three points in line. The constraint matrix has two rows and two columns, both rows are horizontal, and its rank is 1: one freedom left over, pointing straight up, and one dependency among the two bars. Move the joint up and neither bar changes length to first order, which is what the freedom says. To second order both bars get longer, by the same amount and in the same direction, and there is nothing to trade off against — which is what the dependency says. The obstruction is the dependency applied to the second-order stretch and comes to 1.414214; anything but nought there and the freedom is not the beginning of a motion. Lifted by 0.34 the bars are 0.05622 longer, which is the whole argument drawn to scale. Many of one thing

It moves to first order and not at all

Two bars from one joint to two pinned ones, all three in line: the rank leaves a freedom pointing straight up, and lifting the joint stretches both bars. The obstruction is 1.414214, the walk travels a millionth of what it is asked to, and how far it gets is a property of the tolerance rather than of the mechanism.

 The rotor is not a Reuleaux triangle. The generated rotor, with the Reuleaux triangle through the same three apexes drawn over it — three circular arcs, each centred on the opposite corner. They part company by 1.21 mm on a rotor of generating radius 100, and the flank is the one that is right: it is the envelope of the housing, and a Reuleaux flank would foul the wall. The measurement that cannot be argued with is the width. A Reuleaux triangle has the same width in every direction — that is what it is for — and this rotor's runs from 150.0 to 174.1, a spread of 16 per cent. positioned by solving, not by drawing. Drawn wrongly

Six things a shape is not

A rotor that is not a Reuleaux triangle, teeth that are not the same shape as each other, a fillet that is not an arc, a mesh that does not roll, a conjugate pair that cannot be built and a tooth form that was not deduced. Six claims in circulation, each with the number that kills it.

 A ball that remembers the area. Roll a ball round a closed loop on the plane, without ever twisting it about the vertical, and it comes back to the same place turned. The angle is the loop's area divided by r², and the dashed line is that law with nothing fitted to it. The departure at the top is not an integration error: it grows as the square of the angle — a fitted exponent of 1.99 — which is what a leading term's first correction does. The rotation is composed from exact exponentials, so the drawing carries no drift of its own. Wheels, and where they may not go

The ball that remembers where it has been

Roll a ball round a closed loop on a table without ever twisting it, and it comes back to the same place pointing somewhere else. The angle is the loop's area divided by the square of the radius — 0.0016 radians for a 2 mm square under a 50 mm ball — and it is exact in the limit with a departure that is second order in the angle itself.

 The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for. The problem backwards

Where an optimiser starts

An approximate synthesis is a local search on an objective that is non-convex, disconnected and not everywhere defined, so the answer depends on where the search began. Nothing in the optimisation supplies that. What supplies it is the exact constructions the field spent four rungs on, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

 Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept. Linkages

Grashof is a shape test

The oldest classification in the subject compares sums of lengths, so it is unchanged by making the machine bigger — which means a protractor recovers it exactly without recovering a single length. What it does not recover is the margin, and the margin is what says whether the classification is safe.

 Two circles, and an exact straight line. The moving centrode of this motion is a circle of radius 1.5 and the fixed one is a circle of radius 3.0, measured to 8.9e-16. The small circle rolls inside the large one, and a point on its rim traces a diameter of the large one — exactly, with no error term. Watt's linkage is straight to nine per cent of its span and Chebyshev's to twelve; this is straight to 0.0e+0, and the difference is not one of degree. It is the difference between a curve that approximates a line and two centrodes whose rolling produces one. positioned by solving, not by drawing. The motion, not the mechanism

Exact because two circles roll

Watt's straight line is straight to nine parts in a hundred and Chebyshev's to twelve. Here is one that is straight to nothing at all — no error term, no working range, no approximation anywhere — and the reason is that its two centrodes are circles, one rolling inside the other at exactly half its radius.

 Two ways to stop an axle moving sideways. A Panhard rod is one link from the body to the axle, so the axle's end follows an arc and the whole car shifts sideways as the suspension moves: 3.56 mm at 80 mm of travel, and always in the same direction, so it happens twice per bounce. A Watt's linkage keeps the same point on a path that is straight to 33.7 µm — 106 times better, and it is drawn on the same axis, which is why it looks like the zero line. Machines you have met

Holding an axle still

A Panhard rod moves the axle 3.56 mm sideways over 80 mm of travel and a Watt's linkage moves it 34 microns — a hundred times better, and by a higher power. The Panhard's error is quadratic in the travel and the Watt's is fifth order, which is a much stronger statement than "the Watt is better" because it says how the comparison changes with the suspension.

 How many points a linkage will take. function generation: 5 free numbers, 1 net constraint per point, 5 precision points; motion generation: 4 free numbers, 1 net constraint per point, 5 precision points; path generation: 9 free numbers, 1 net constraint per point, 9 precision points. Every count in the literature that differs from these differs about what is being counted as free, not about the geometry. The problem backwards

How many points may be prescribed

Five poses, five angle pairs, nine points — three numbers that get quoted as properties of a four-bar and are properties of what somebody decided to count as free. Derive them instead, and the fifth precision point turns a linear solve into a system with 128 paths, twelve finite solutions, four real ones, and exactly one linkage anybody could build.

 One tendon over two joints. A two-link chain with a strand anchored off to the left, running over an idler at each joint and terminating on the far link. Each idler is centred on its joint's axis, which is the one arrangement that makes the strand's length a linear function of the joint angles: the coupling measured here is 14.000000 and 10.000000 mm per radian, against radii of 14 and 10. Total strand 232.845 mm. positioned by solving, not by drawing. Members that pull

One strand over many joints

Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.

 The number on the tackle is a limit. The velocity ratio of a 4-part tackle as its blocks separate, measured by differentiating the strand's own length. It is 3.6174 at 90 mm and 3.99992 at 6898 mm, and it reaches 4 nowhere. The shortfall is geometric and has nothing to do with friction: each part of line makes an angle with the lift, and a part at angle β shortens by cos β of the movement. A tackle used at close quarters — which is when a tackle is useful — is the case furthest from its own rating. Drawn wrongly

Six things a strand is not

A tackle that is not four to one, a wrap that is not a half turn, a tensioner whose travel takes up nothing, a winch speed that is not a property of the winch, a cable rig that holds nothing still, and a shaped pulley that cannot be asked for what it is usually asked for. Six claims, each with the number that kills it.

 The search generates 3,000 candidates for 1,878 answers. How much work the enumeration does, against how much it has to show for it. The upper line is the number of complete labelled graphs the search reaches and the lower is the number of distinct graphs they turn out to be, so the vertical gap is waste — every candidate above the lower line is a graph the search had already found under a different labelling. At eight links the unpruned version of this search generated 8,494 candidates for the same 71 answers, and at ten links it did not finish at all; with the pruning it generates 3,000 for 1,878 in 442 milliseconds. The rule that does it is one line long: when two links carry the same number of pins, reject the labelling that would be lexicographically smaller if they were swapped. It cannot reject a labelling that is the largest in its class, so nothing is lost, and it is not a complete test, which is why the canonical form is still taken at the end. The chain before the lengths

The candidates a search throws away

The obvious enumeration generates every labelling of every chain and keeps one. At eight links that is 8,494 complete graphs for 71 answers; at ten it does not finish. One rule — reject the labelling that a swap of two equal links would improve — takes it to 3,000 candidates for 1,878 answers in half a second, and twelve links is still out of reach.

 How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor. Wheels, and where they may not go

The wheel that forbids nothing

Every wheel contributes the same row to the same matrix, and whether that row is a constraint on the vehicle or a statement about the wheel's own speed is decided by one factor of sin γ. At γ = 0 the vehicle may not move across the wheel; at 45° the row says nothing about the vehicle at all, and sideways costs exactly what forwards costs — to the last digit, and at no other angle.

 Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be. More than one input

Three mechanisms, one subtraction

A micrometer's differential screw, a chain hoist's differential pulley and a robot joint's compound epicyclic look nothing like each other and are the same device. Each takes two nearly equal quantities and returns their difference, each buys its enormous ratio with that difference, and each carries the same conditioning number — |a/(a−b)| — measured here by perturbing the mechanisms rather than by quoting the formula.

 Sarrus, as two planes meeting in a line. Each arm of Sarrus's linkage is three pins with parallel axes, so each arm holds the platform inside a planar group — the one whose normal is that arm's axis direction. The platform has to satisfy both, so what it may do is the intersection, and the intersection of two planar groups whose normals are not parallel is the one-dimensional group of translations along their common perpendicular. The platform goes up and down and does nothing else, and that is the exact straight line the spatial field measured to 10⁻¹⁶ of its span — arrived at here with no mechanism solved and no tolerance anywhere. Drag the arms towards each other: the answer is a translation at every angle but zero, where the two groups become one group and the intersection jumps to three dimensions. That is the linkage built flat, and it is the configuration in which it stops being a straight-line mechanism. What a joint is

Two planes meeting in a line

Sarrus's linkage draws an exact straight line out of six pin joints, and the spatial field proved it by solving the mechanism sixty times and measuring a departure of 9.8 × 10⁻¹⁶. Here the same fact comes out of two planes and a cross product, with no mechanism solved anywhere — and the two routes are not redundant, because only one of them can tell you the linkage as built delivers it.

 Which teeth one tooth ever meets: 20 on 40. Follow one tooth of the pinion round and mark every wheel tooth it touches. It does not touch them all. It touches 2 of 40, which is z₂ divided by the greatest common divisor of the two counts — 20 here — and it goes on touching the same ones for as long as the gears are in mesh. The pattern repeats after 2 turns of the pinion. Adding one tooth to the pinion makes the counts coprime and takes the count from 2 to 40. The marks are produced by walking the mesh, and the count they give is compared with the gcd rather than derived from it. Teeth

Which tooth meets which

A tooth on a pinion does not meet every tooth on its wheel. It meets z₂ divided by the greatest common divisor of the two counts, and it meets the same ones for the whole life of the drive. On the default planetary used throughout — sun 24, planets 24 — a planet tooth touches exactly one sun tooth and never touches another, and nothing in the drawing says so.

 20 teeth driving 32. Both flanks generated from the involute, not approximated, at a pressure angle of 20°. The orange line is the line of action — tangent to both base circles, and the only place contact happens. Its length between the two tip circles divided by the base pitch is the contact ratio, 1.612 here, which means that for 61% of the cycle two tooth pairs are carrying the load and for the rest just one. The velocity ratio is 0.6250, and it is constant because the common normal never moves. The dashed extension runs between the two base tangency points, which are 8.89 mm apart; the heavy part is where contact actually happens. Teeth

The module is a size, the ratio is a shape

A spur pair has exactly one length in it. Every quantity it produces is either proportional to that length or completely independent of it — centre distance, base pitch and contact length scale exactly; ratio and contact ratio do not move at all, not even by a part in ten to the fifteen.

 A flat face on a base circle of 30, where the profile bends tightest. The cycloidal programme — rise 10 over 120°, return over 120° — cut for a flat-faced follower on a base circle of 30, turned to 212.4°, where the profile's radius of curvature is smallest. The contact sits −5.37 from the follower's axis, which is ds/dθ, and the dashed circle is the one that fits the profile there: R₀ + s + s″ = 24.668, and 24.668 fitted through the drawn curve. The face's pressure angle is zero at every angle. What it needs instead is for that radius to stay positive, and it does so on every base circle above 5.332. Prescribed motion

A flat face asks for a convex cam

A flat-faced follower is pushed at right angles to its face, so its pressure angle is zero at every angle and the constraint that sizes a roller's cam disappears. What replaces it is convexity: the profile's radius at the contact is R₀ + s + s″, and a base circle below 5.332 leaves a cycloidal cam with a hollow the face cannot reach — held 0.446 high on a base of 2.

 One crank, four machines, one area. Four crank-rockers sharing only a crank of length 1, with grounds, couplers and rockers all different, each tracing the point 30% of the way from the crank pin to the rocker pin. The curves have different shapes and sizes and different places in the plane, and the area each one encloses is 2.199115 = (1 − 0.3)·π·1² — on both assemblies of every one, with a worst difference of 3.2 × 10⁻¹⁴. The paths points trace

The area a coupler point encloses

Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.

 11 assortments are arithmetically possible and 7 contain a mechanism. The 10-link census organised the way every published table organises it: by how many links carry two pins, three, four and more. The assortments themselves are a small piece of arithmetic — the degrees must sum to twice the pin count and none may be below two — and it admits 11 of them. 4 contain no chain at all. Each of those 4 needs a link carrying six, seven or eight pins, and a link with that many pins in a chain this small always drags a structure in with it: the graphs exist, they satisfy Grübler exactly, and every one of them has a rigid subchain. That is a result the arithmetic cannot reach, because the arithmetic never looks at where a pin goes. The chain before the lengths

Eleven assortments and four that are empty

How many links carry two pins, how many carry three, how many carry four: two lines of arithmetic admit eleven answers at ten links. Seventy-eight graphs have degrees the last four of them describe, every one of those graphs satisfies Grübler's rule exactly, and not one of them is a mechanism.

 Every axis through one point. Four wheels on one rigid body, each rolling without sliding. Each turns about some point on its own axle line, and a rigid body has one such point, so every axle line has to pass through it. That is the whole of steering geometry, and it is a rank condition rather than a formula: here the four rows have rank 2 of 3, leaving a one-dimensional family of twists, and the centre they agree on is 12.000 m to the side. The scrub is 3.2e-17 m per metre — zero, to the last digit. Wheels, and where they may not go

Every axis through one point

Bolt several rolling wheels to one rigid body and they impose one condition between them: every axle line must pass through a single point. The familiar steering formula falls out of it as a consequence rather than being quoted — cot δₒ − cot δᵢ = 0.574074 at a turn of six metres, of eight, of twelve and of twenty, on a track of 1.55 m and a wheelbase of 2.7.

 In space the arithmetic allows almost nothing. A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n − 7)/5 joints — and that is an integer only when the link count leaves a remainder of two on division by five. The whole table is this: 7, 12, 17, 22 links, and nothing else. At seven links the degrees must sum to fourteen across seven links with none below two, so every link is binary and the graph is a single seven-cycle: there is exactly one spatial chain, and it is a loop. That is the census explanation for something the spatial field has lived with since it was written — every spatial mechanism on this site is one closed loop — and it had never been stated as a count. The next admissible size is twelve links and thirteen joints, where two assortments are arithmetically possible, 157 candidates give 33 graphs, and 5 of them are chains — every one with ten binary links and two ternary, so the assortment with a quaternary link is empty exactly as four of the planar ones are. Out of the plane

In space there is one chain

A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n−7)/5 joints — an integer only when the link count leaves a remainder of two on division by five. At seven links every link is binary, the graph is a single seven-cycle, and there is exactly one spatial chain.

 Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion. Drawn wrongly

Six things a network is not

A count that is right about a difference and read as an answer, a nullity taken for a mobility, a flat state that cannot tell a mechanism from a structure, a scissor ring that closes nowhere, a vertex that folds while its sheet does not, and a null space computed with an instrument whose floor is above the answer. Six claims, each with the number that kills it.

 53/11, instant by instant. The ratio a 53-tooth sprocket and a 11-tooth sprocket actually deliver, against the driver's rotation. The quoted 4.818 is the flat line, and it is exactly right as an average — the mean over a turn comes out at 4.8182, which the computation is never told. Within every tooth the ratio swings from 4.756 to 4.948, a fluctuation of 3.99%. The sharp corners are pins seating: the identity of the pin the strand runs from changes 53 times a turn on one sprocket and 11 times on the other, and the two do not coincide. Machines you have met

The chain is a polygon

A chain's pins sit on a polygon, so the radius that matters swings by 1 − cos(π/n) within every tooth: four per cent on an eleven-tooth sprocket. The quoted 53/11 is the mean of that, exactly — and how much of the fluctuation reaches the back wheel is decided by the fractional number of pitches in the taut strand, which nobody adjusts and which can take the drive from perfectly uniform to two per cent.

 Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be. More than one input

The reductions a planetary cannot give

One epicyclic offers six ratios, and the formula for each of them suggests the whole positive line is available. Sweep every design that can actually be cut and assembled and the reachable set has a hole in it running from 1.630 to 2.586 — the width of which has a closed form — and a reduction of exactly 2, the most ordinary thing anybody asks a gearbox for, sits in the middle of it.

 Two taut paths, and no way between them. A strand from one point to another past a peg. Each side gives a path that is taut — straight where it can be, on the surface where it must be, leaving at a right angle — and each is the shortest path on its own side: 219.165 mm on one and 200.643 mm on the other, against 200 mm of open air the strand cannot use. Neither can turn into the other without passing through the peg, which is a different kind of non-uniqueness from the assembly branches of a linkage: those are separate roots of one equation, and these are separate classes of one minimisation. positioned by solving, not by drawing. Members that pull

The taut path has more than one answer

A strand from one point to another past a peg has two taut paths — 200.643 mm on one side and 219.165 on the other, against 200 mm of open air it cannot use. Both are shortest. Neither can become the other without passing through the peg, and no computation recovers which one was threaded.

 What the repeated constraints cost the drawing. Move an interior vertex of the flat pattern and the folded state generally stops existing. It survives if the change to the vertex closures can be absorbed by a change in the fold angles — and the part that cannot be absorbed is exactly the part that lies along a dependency, because a dependency is a direction in residual space the fold angles cannot reach. So the number of conditions a pattern's shape has to satisfy is at most the number of dependencies among its constraints, and on the Miura family it is exactly that: one at three by three, four at four, nine at five, measured by taking the rank of the obstruction. A twelve-by-twelve sheet has a hundred conditions on where its vertices may be. That is why a grid whose vertices are anywhere at all does not fold, and it is the same number, read the other way round, as the amount by which the count is wrong. Many of one thing

What a pattern has to satisfy

Move an interior vertex of a crease pattern and the folded state generally stops existing. How many conditions the drawing has to meet is not a matter of taste: it is exactly the number of dependencies among the constraints, measured at one, four and nine on three sizes of sheet, and a hundred on a sheet of a hundred and forty-four panels.

 What a singularity does, and what it does not do. The reflector driven straight through the configuration at which its two placements merge — here θ = 0.800, where the rhombus flattens onto its own mirror. Two numbers are plotted. The closure residual is how well the bars are satisfied, and it does not move: 8.3e-14 on both sides. The departure is how far the output is from the angle the gadget is supposed to produce, and it goes from the floor to order one at 0.800. Nothing breaks. The gadget goes on being a perfectly good linkage and stops being the function it was built to be. The curve as an equation

Where the machine stops being the function

Drive a reflector through the angle at which its rhombus flattens and it comes out computing something else. Nothing breaks: every bar is the length it was, the closure residual stays at 8 × 10⁻¹⁴, and the machine goes on turning. That is why every compiled machine in this field works over an arc and not a turn — the quintic's over a tenth of a radian.

 Two routes to the same derivative. How much the output angle moves when the coupler length moves, through one turn, computed twice. One route rebuilds the mechanism at b ± 10⁻⁶ and solves both from scratch; the other differentiates the constraint equations and solves one linear system against the analytic Jacobian. They lie on top of each other — the strip beneath plots the difference on a four-decade log scale, and its largest value anywhere in the turn is 2.9e-10 against a sensitivity of order 0.41. That is the only independent check there is of the Jacobian itself, whose coupler rows carried four wrong signs from the foundation phase to 2026-08-12 without ever drawing anything wrong. As built

A band with a direction in it

One whole direction of a four-bar's tolerance box does nothing. A machine made a quarter of a per cent too big all over has an output error of exactly zero — and an aluminium four-bar heated by a hundred degrees has an output error of exactly zero, while one with a steel frame has 0.076°.

 The circle of points going straight, at 66°. Every point on this circle is, at this instant, travelling in a straight line: its path has zero curvature there. The circle passes through the pole — where the point is not moving at all — and its diameter is 47.64, which is the pole's own speed divided by the plane's angular rate. Nothing here was assumed to be a circle. The locus is the zero set of a quadratic whose |w|² coefficient is φ′³, a real number with no cross term and no difference between its two square terms, and a general conic fitted to the sampled locus returns those coefficients at 8.1e-15 and 6.4e-15. At this position the coupler is close to translating, the pole has run off the canvas and the circle with it — the figure is the size of the mechanism, and δ here is 13.6 coupler lengths. positioned by solving, not by drawing. The motion, not the mechanism

A curvature is a size with a minus sign

Everything in the curvature field is a similarity invariant in shape and a reciprocal length in value. Scale a moving plane and its inflection circle scales, its cubic of stationary curvature scales, and every curvature it computes is divided by the factor — so a bigger machine traces gentler paths than its drawing suggests.

 Every length wrong, every reading right. A four-bar was built to the dimensions in the upper bar of each pair and its output angle read at 30 positions. A calibration started from the nominal dimensions returns the lower bar. It reproduces every one of those readings to 1.81e-16 radians and not one of its four numbers is the machine's: they are the machine's multiplied by 0.992289, every one of them, to 3.0e-16. The shape is recovered exactly — the distance in Freudenstein's three invariants is 6.3e-16 — and the size is a free parameter the damping happened to leave near where it started. A machinist handed these numbers would build a machine that works and is not this one. Numbers that were measured

Every length wrong, every reading right

A four-bar was built out of true and measured at thirty positions. A calibration started from the nominal dimensions reproduces every reading to 1.8 × 10⁻¹⁶ radians and returns four lengths, not one of which is the machine's. They are the machine's, multiplied by 0.99229 — every one of them, to fifteen figures.

 The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine. More than one input

The transmission angle has no size

The geometric half of force transmission is an angle in a triangle whose three sides scale together, so it is the same at every size — 54.31° at its worst on this machine, whatever units the drawing is in. Which means a measurement that cannot recover a single length recovers the whole of what this field computes.

 Every body of the Stephenson six-bar, sliced by one line. One random complex line, and a point on each body of the Stephenson six-bar required to lie on it. Every system has Bézout number 32. crank: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; ternary coupler: 12 configurations drawing 6 distinct points, each 2 times, so degree 6; rocker: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; arm: 18 configurations drawing 18 distinct points, each 1 time, so degree 18; output: 12 configurations drawing 2 distinct points, each 6 times, so degree 2. The configurations are what the tracker counts; the points are what the curve has. How many answers

The curve nobody eliminates

A point on the arm of the site's dwell six-bar draws a curve whose equation nobody writes down and no fit can find: at degree eighteen a fit needs a hundred and ninety coefficients, and its singular values have no gap to decide by. Sliced by a random line, the same machine has thirty-two paths to track and eighteen arrive, on every line tried, all eighteen distinct and all on one curve.

 Every pivot on these arcs gives a 60° swing at a time ratio of 1.2. The rocker's pivot O₄ is fixed and its two limit positions B₁ and B₂ are 60° apart. At a limit the crank and coupler are in line, so the crank pivot sees the chord B₁B₂ at the angle δ = 180°(Q − 1)/(Q + 1) = 16.36°, and the points that see a chord at a fixed angle are arcs of two circles. The thick stretches are the 828 sampled pivots that give a crank-rocker with exactly this swing and ratio; the rest of each circle gives a linkage of another kind. The linkage drawn is the member whose worst transmission angle is largest, at both limits: ground 1.222, crank 0.478, coupler 1.130 and rocker 1, with its worst transmission angle 40.32°. Linkages

A swing and a time ratio

A shaper's specification gives the rocker's swing and how much quicker the return must be than the cut, and those two numbers do not fix a linkage. They leave a one-parameter family of crank-rockers on the arcs of two circles, every member exactly right, and the transmission angle chooses between them — which is also what decides that a 60° swing cannot return more than 1.207 times as fast and keep 40°.

 Three machines drawing one oval, and where each keeps its area. The crank-rocker with ground 4, crank 1, coupler 3.5 and rocker 3, its tracing point at (0.45, 0.5) of the coupler, and the two other four-bars Roberts's construction gives for the same curve, each drawn holding the same point of one oval and each with its input pin's path dashed in the input colour and its output pin's in the output colour. The shaded oval encloses 2.116354 for all three. In the crank-rocker the crank pin goes round and carries 1.727876 of it; in the double rocker neither pin goes round and the coupler's turn carries 1.727876; in the rocker-crank the output pin goes round and carries 1.727876. The remaining 0.388478 is the same in all three. The paths points trace

Where three machines keep one area

Roberts's theorem gives every four-bar two others that draw the same coupler curve, and so enclose the same areas. Measured, they do, to 10⁻¹² — but each keeps the area in a different place: the crank-rocker in its crank pin's circle, the double rocker in its coupler's turn, the third machine in its output pin's circle. What is left over has no closed form, and it is one number in all three.

 A catalogue is a search space, and a requirement is a filter on it. What a census is for. Four requirements applied in turn to the 230 ten-link chains, each of them a statement about the graph alone: a link carrying four pins, a link none of whose neighbours is binary, and a way of driving it that comes apart into dyads. 26 chains survive all of them. None of this is dimensional synthesis and none of it can be — no requirement here mentions a length, an angle or a position, and every one of them can be checked before a single dimension is chosen. That is the argument for having the census at all: the design problem is a search over shapes within a topology, and knowing which topologies there are turns an open question into 26 closed ones. The chain before the lengths

A catalogue is a search space

Dimensional synthesis searches over lengths within a topology, and the topology is chosen first — usually from memory, usually from a list of five. With a census the list is two hundred and thirty, every requirement that reads only the graph is a filter on it, and the choice stops being a habit.

 A reduction made out of a difference. A compound epicyclic's reduction is 1/(1 − z₁z₄/z₂z₃) — a ratio of two integers, so it is exactly what a catalogue says, at every position, forever. It is the only number in this field that survives being measured, and the reason is that it is a count. What it is not is insensitive: each row's denominator is a difference of two nearly equal products, and the last two rows differ by one tooth on one ring. Machines you have met

A ratio that is a count

A compound epicyclic's reduction is a ratio of two integers, because it is a ratio of tooth counts — 2176/106 for the drive here, exact at every position and on every unit ever made. It is the only quoted number in this field that survives being measured, and it is fragile in a way exactness does not protect against: one tooth on one ring moves it by forty per cent.

 Exactness is not bought with links. Five straight-line mechanisms, each measured over its own working arc — walked out to its dead centres and back a tenth — and each plotted at its own bar count. Watt's four bars are wrong by 9.0 per cent of the stroke and Chebyshev's by 12.4; Peaucellier's seven are exact. There is nothing in between, and adding bars to an approximation does not walk down the axis: the compiled machine is exact for the same reason Peaucellier is — an exact algebraic relation — and its extra bars buy generality rather than accuracy. The curve as an equation

Exact costs more than close

Watt's four bars are wrong by nine per cent of their stroke and Chebyshev's by twelve. Peaucellier's seven are exact to 4 × 10⁻¹⁶, and a compiled machine is exact to 4.8 × 10⁻¹⁴ in five. There is nothing in between — adding bars to an approximation does not walk down the axis, and the four-bar in every beam engine ever built is on the wrong end of it.

 The pose set is finite exactly when the part is held. Two arrangements, both with 0.06 of clearance on every contact, with the set of positions the part's centre may occupy drawn to scale. The one on the left holds: its pose set is a small bounded polyhedron, and every dimension of it is proportional to the clearance. The one on the right does not: its pose set runs off the page in the direction the part slides out of the vee, and giving the contacts a tighter tolerance narrows the box without ever closing that direction. A tolerance cannot buy a hold. The clearance decides how big a finite pose set is and the arrangement decides whether it is finite, and the second question has to be settled first because no amount of the first will settle it. Contacts that only push

Held is not located

Back every obstacle off by a clearance and the permitted poses become a polyhedron — bounded exactly when the arrangement is a hold, since an unbounded direction of it would be a ray of the escape cone. So whether a part is held is whether its pose set is finite, the clearance is what gives that set a size, and the two questions have to be settled in that order because no tolerance settles the first.

 Three points, three orders of contact. Replace the coupler point by a crank pivoting at the centre of its path's osculating circle, drive the linkage away from the instant, and measure how far the point gets from that circle. The distance grows like a power of the step, and the power is the number of derivatives that agreed. An ordinary point gives 3.00; a point of the cubic of stationary curvature gives 3.94; a Burmester point, where the curvature is stationary and its rate of change is too, gives 4.97. Nothing in the measurement knows which kind of point it was handed. This is what the two special curves are for: they are where a single pivot can replace a whole linkage for longest. positioned by solving, not by drawing. The motion, not the mechanism

How long a pivot stands in for a linkage

Throw the four-bar away and replace a coupler point by a crank pivoting at the centre of its path's curvature. The two separate like the cube of the crank step for an ordinary point, the fourth power on the cubic, and the fifth at a Burmester point — three exponents that are the whole field, measured with no derivative anywhere in the measurement.

 What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14. The problem backwards

Prescribing a curve rather than points

A four-bar can be made to pass through nine prescribed points and no more; past nine the problem is over-determined and the answer is an optimiser's. Prescribe the whole curve as an equation instead and there is no counting to do — but the mechanism that comes back has four hundred bars where the four-bar had four.

 Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing. Drawn wrongly

Six things a hold is not

A rank read as a restraint, a count read as an answer, four contacts placed the wrong way round, a nullity taken for a spin, a part free in every direction and unable to leave, and a tolerance offered as a cure for an arrangement that was never a hold. Six claims, each with the number that kills it.

 Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it. What a joint is

The instrument that is not a derivative

Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.

 Reverted trains of 12 : 1. A reverted train has its input and output shafts in line, which means the two stages share one centre distance — so the tooth counts must satisfy z₁ + z₂ = z₃ + z₄ and give the ratio asked for. Two equations in four integers, and there is no reason for a solution to exist. For 12 : 1 there is none at all until the wheels are allowed to reach 63 teeth; below that the coaxial condition and the ratio simply cannot both be met. This is the same kind of arithmetic as the clock trains of the timing field, with one extra equation, and the extra equation is what a shaft position costs. More than one input

Two shafts that must be in line

Asking a two-stage gear train for a ratio is easy. Asking it for a ratio and for its input and output shafts to be coaxial is asking for a solution of two equations in four integers, and there is no reason for one to exist. A twelve-to-one reverted train needs a sixty-three-tooth wheel before it has any solution at all — while sixteen to one, a larger ratio, manages with fifty-six.

 Half a tooth, spent three ways. Each bar is one beat of the escape wheel: exactly half a tooth pitch, 6.0° on 30 teeth, whatever the faces are cut like. The dark part is the impulse, which is the only part that does anything to the pendulum; the pale part is the drop, in which nothing is touching anything; the short tail is the lock-in run, in which the arriving tooth drags the wheel backwards as it settles. On the arc with no draw that tail is exactly zero and the budget has two terms. On every other face it is not, and the three still sum to the half pitch to twelve figures — which is the check, since the three are computed from three different contacts. As built

What a drop cannot be smaller than

Two thirds of an escape wheel's travel is drop, and drop does nothing. The obvious economy is to cut it down, and it cannot be cut down, because every dimension it is made of has a tolerance and a drop smaller than the accumulated error is a tooth that does not clear the pallet it is leaving. The stack is 0.39°, and it barely moves when the tooth count triples.

 Every joint's lever arm. The thin lines run from the tool to each joint's axis, meeting it square. Their lengths are what a radian of error at each joint costs the tool in metres — not a rule of thumb but the Jacobian column, which is ω × r and therefore that perpendicular exactly. The shortest of them belongs to the joint nearest the work and the longest to the joint furthest from it, which is why an arm's accuracy is decided at the shoulder and its resolution at the wrist. One path to the tool

Where an error at the shoulder ends up

The same angular error at every joint of an arm, and the tool is out by 0.156 mm because of the shoulder, 0.017 mm because of the wrist roll and exactly nothing because of the last joint. The numbers are not properties of the joints. Each one is the distance from the tool to that joint's axis, measurable off the drawing with a ruler.

 What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing. Members that pull

Where a strand stops touching

A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.

 The ratio survives; the continuity does not. An involute pair holds its ratio at any centre distance, and that is not the same as working at any centre distance. The contact ratio — the length of the contact path divided by the base pitch, which counts how many pairs of teeth are engaged at once — starts at 1.647 for this 24 : 36 pair and falls as the shafts move apart, because the useful part of the line of action is bounded by the two tip circles. It reaches one at 2.82 mm, and below one a pair of teeth lets go before the next has picked up: the drive stops being continuous and becomes a series of arrivals. That is the real limit on the involute's indifference, and it is a limit on the teeth rather than on the tooth form. The shape is the unknown

Where two shapes stop touching

A conjugate pair is exact at every instant it has a contact. It does not have one for ever: a profile is an arc rather than a curve, and both ends of that arc are somebody's decision — which is why the useful question about a pair of shapes is not whether they mesh but for how long.

 The instrument's error, multiplied. The error in the recovered shape against the error in each reading, over four decades, each point the mean of six independent calibrations and the open marks the worst of the six. The slope is 0.9994 — the error is linear in the noise, with no threshold and no saturation — and the constant is 1.90. So a protractor good to a milliradian gives a shape good to about 1.9 milliradians' worth, and the factor belongs to the mechanism and the poses rather than to the instrument. The bound from the smallest singular value is 2.16, which the measurement sits under, as it must. Numbers that were measured

The instrument's error, multiplied

Repeat a whole calibration on independently noised readings at four levels three decades apart and the error in the recovered shape is linear in the noise, with a fitted slope of 0.9994 and a constant of 1.90. That constant belongs to the mechanism and the poses, not to the instrument — and it is bounded by one over the smallest singular value.

 Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off. The problem backwards

What a synthesis assumes it knows

Every construction in this field is handed a demand in absolute coordinates — three positions of a coupler plane, at stated places — and returns a linkage in the same coordinates. Scale the demand and the answer scales, which means the construction's whole content is about shape and its answer carries a size it was given.

 A geared five-bar at 3 : 2, its coupler pin traced over 2 input turns. Two cranks of length 1 on pivots 3 apart, meshed through gears whose pitch circles, dashed, have radii 1.800 and 1.200, so the second crank turns 1.5000 times for every turn of the first and the other way. Coupler bars of 3 and 2.5 join them at the pin C, and the thin curve is where C goes over 2 turns of the input. The two crank pins are always between 1 and 5 apart and the bars can span 0.5 to 5.5, so the chain never meets a dead position. After 2 turns the machine is exactly where it began and the curve is closed. What can move

One freedom, and a motion that never repeats

Mesh a gear on each crank of a five-bar and the count and the rank agree that one freedom is left, at every gear ratio. Whether the machine ever comes back to where it started is a different question, and neither instrument can see it: at 3 to 2 it is home after two turns, at 37 to 23 after twenty-three, and at the golden ratio never, with its nearest returns at the Fibonacci numbers.

 A dwell six-bar built on the vertex of a symmetric coupler curve. The four-bar with ground 3, crank 1, and coupler, rocker and arm all 2.5, with the angle at the rocker pin set to 120.8024°, so that its coupler curve is its own mirror image about the dashed line. The curve crosses that line at crank angle 0°, and a link of the osculating radius there, 5.7691, runs from the tracing point to a pin at the centre of curvature; an output link of 3 from a third ground pivot holds that pin, square to the dwell link at the vertex. Faintly, the dwell link and output 50° of crank either side. The output swings 7.43° over a whole turn, and near the vertex its angle changes only at sixth order in the crank's. The paths points trace

The flattest dwell is not the longest

A coupler curve that is its own mirror image has no odd terms in its distance from a circle centred on the mirror line, so one angle of the coupler can remove the fourth-order term and leave a dwell of sixth order, with no search of the curve. A six-bar built there dwells for 58.7° of crank inside 0.1% of its swing. Turned two degrees away from that angle it dwells for 80.1°, and the searched six-bar for 24.4°.

 What a pin's size costs a synthesis. Every one of the 1,176 exactly correct three-position syntheses the synthesis field's survey produces, asked a question its three verdicts cannot ask: are any two of its pins closer together than the material round them? The poses span about two units, so the horizontal axis is a pin as a fraction of the mechanism. At a boss radius of 0.05 nothing is lost; at 0.4 — a pin nearly a fifth of the pose span — 27% of the exact solutions cannot be built, and 15% of the ones that had already passed the branch, circuit and order tests go with them. The defect is real, it is rarer than the kinematic ones, and it is not correlated with them. The problem backwards

A defect that is not kinematic

The synthesis field's survey ends in three verdicts and all three are about which solutions a sweep visits. Here is a fourth, found by asking how much room two pins need — and it is the first defect on this site that a simulation cannot find, because in the equations a pin is a point.

 Two permitted motions, and a composite that lifts off. A disc resting on a straight edge — a roller follower on a flat-faced cam, and the simplest higher pair there is. Two bodies touching at a point rather than over a surface, two freedoms: slide along the edge, and turn, because a disc is its own symmetry group about its centre. Both are permitted and both keep the contact exactly. Their composite does not. The faint discs are the two permitted displacements taken separately; the solid one is one followed by the other, and its centre sits 1.049 radii off the dashed line where a tangent disc's centre has to be. The excursion is exactly |t₂ sin φ₁| — the second displacement's slide times the sine of the first one's turn — derived from the two displacements rather than from the composition and agreeing to 10⁻¹⁶. A lower pair's freedoms compose and a higher pair's do not, which is why a joint's freedom count is the dimension of a group in one case and the dimension of nothing in the other. What a joint is

A higher pair has no group

A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.

 A tooth flank is the end of an unwound strand. A strand wrapped on a circle of radius 45.105 mm, unwound while kept taut. Its free end traces the involute — the same curve the gears field builds from its own parameterisation, agreeing to 1.5e-14 mm over the whole flank. The strand is the important part of that sentence and not the curve: the taut portion is a tangent to the base circle, its length is the arc it has left, and both of those are statements about a strand rather than about a tooth. At this position the free length is 27.965 mm. positioned by solving, not by drawing. Teeth

A tooth flank is an unwound strand

Unwind a taut string from a circle and its free end traces the involute, to 1.5·10⁻¹⁴ mm of the curve the gears field draws. That is not a curiosity: the line of action of an involute pair is a crossed strand on the two base circles, so the property the involute is chosen for — a ratio that does not care where the shafts are — is a belt's property rather than a curve's.

 One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry. Drawn wrongly

Six things a chain is not

A count read as a verdict, a rank trusted where it is blind, a fingerprint used as a proof, a list of five taken for a complete one, a solver treated as a convenience, and a census read as a catalogue of machines. Six claims, each with the number that kills it.

 How far the driven link turns is a fact about the lengths, not the chain. Each of the sixteen eight-link chains, given the arbitrary placement its own layout produces, driven from its first available choice of frame and input, and swept until a frame stops closing. 10 of the sixteen reach every angle and the rest rock through between 107° and 244°. Nothing in this chart is a property of the chains. Change the placement and the bars change; the census above them does not. It is here because it is the sharpest way to say what this field does and does not decide, and because the temptation to read a topology census as a catalogue of machines is exactly the mistake it prevents. The chain before the lengths

The chain has no lengths

Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. Which is also the statement of what a census cannot decide — and the sixteen eight-link chains, each given one arbitrary set of dimensions and driven, produce a chart in which nothing belongs to the chains.

 Bars against terms, over the whole catalogue. One mark per compiled machine. The bar count rises much faster than the term count, and the reason is the summing chain: term k has to have its direction carried to the k−1th vertex of the chain, one parallelogram per hop, so the carrying costs a translator for every pair of terms. Nine curves, from five bars to four hundred and thirteen, on a term count that goes from one to eighteen. The curve as an equation

What universality is worth

The linkage exists, it is four hundred and thirteen bars, and it draws ten degrees of its curve. All three are true and only the first is in the theorem — which is the ordinary shape of a result about what exists, and the reason it was worth building one to find out.

 An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing. Members that pull

A drum is a size, a wrap is a shape

A strand's whole behaviour is decided by where it leaves each body and how far round it goes, and both are angles. So a belt drive's velocity ratio, its wrap angles and its tackle's mechanical advantage transfer between drives of any size — and the one thing that does not is how much strand there is.

 A parallelogram at its flat position, exact and built slightly wrong. Left, a parallelogram with ground 3, cranks 2 and coupler 3 lying flat, where its two assemblies meet; faintly, the two ways it can go on, drawn at 30°. Middle, the same linkage with its coupler 0.05 too long, drawn at the closest it can come to the flat position: 7.49° away on either side, so the input's circle is thick where the input can go and red across the 15.0° it can never enter. Right, the input crank 0.05 too short, at the flat position: its two assemblies put the output crank 43.76° apart, and they do not meet at any input angle. What can move

A parallelogram a micron wrong

A parallelogram linkage sits exactly where two kinds of four-bar meet, so a parallelogram that has actually been made is always one of four other machines. Make one bar a micron wrong on a 300 mm frame and the input stops a tenth of a degree short of lying flat, or the output turns round there with an acceleration that grows as one over the square root of the error. A third bar turns the square root back into a misfit of one micron.

 Where a fit's singular values fall away, on each motion of a parallelogram. The 28 singular values of the degree-six fit, largest first and relative to the largest, for points traced on the parallelogram's circle motion, on its quartic motion, and on both. On the circle motion the last 15 lie below a drop of 3.7 × 10¹³, from 0.232 to 6.2 × 10⁻¹⁵; on the quartic motion the last 6 lie below a drop of 3.0 × 10¹⁰, from 6.3 × 10⁻⁵ to 2.1 × 10⁻¹⁵; on both motions the last one lies below a drop of 2.5 × 10¹², from 3.1 × 10⁻⁴ to 1.2 × 10⁻¹⁶. A drop of ten decades or more is a null space that is exactly there: fifteen sextics vanish on a circle, six on a quartic, and one on both. The paths points trace

A null space of fifteen is not noise

Points traced on one motion of a parallelogram four-bar leave a degree-six fit with fifteen polynomials that vanish on them, behind a drop of thirteen decades: the circle times every quartic. Half an oval of an ordinary coupler curve leaves two to five, behind drops of two. The count is the same kind of number in both cases, and only the drop beside it says which one is algebra.

 The closed path of a geared five-bar's pin at four gear ratios. A five-bar with two cranks of length 1 on pivots 3 apart, couplers 3 and 2.5, and a gear on each crank, drawn over one whole cycle at ratios 1 : 1, which closes after 1 input turn; 2 : 1, which closes after 1 input turn; 3 : 2, which closes after 2 input turns; 5 : 3, which closes after 3 input turns. Both assembly branches of the coupler pin are drawn, one in each colour. The curves they make together have degree 6, 10, 16, 26. How many answers

Every rational gear ratio has a degree

Mesh a gear on each crank of a five-bar and the pin where its couplers meet draws a closed curve at every rational ratio. Its degree is 6 at 1 to 1, 16 at 3 to 2, 68 at 13 to 8 and 178 at 34 to 21: four times the larger term of the ratio plus twice the smaller, read off the span of one polynomial in one variable. Along the golden ratio's convergents the degree grows by the golden ratio at each step, and at the golden ratio itself no fit finds any.

 One pin and two, on the same wheel. A 6-slot Geneva wheel at one instant of its index, driven by a crank carrying one pin and by the same crank carrying two. The wheel, the slots and the centre distance are identical; only the pin count differs. Each pin drives the wheel through one slot pitch while the driver sweeps 120°, so 1 pin gives 1 index a driver turn and leaves 67% of it at rest, 2 pins give 2 indexes a driver turn and leave 33% of it at rest. Nothing else about the mechanism changes, which is why the whole question is arithmetic. Motion that stops

Two pins and no dwell at all

Put a second pin on a Geneva's crank and the wheel indexes twice a turn instead of once, at a quarter of the acceleration for the same output rate. Put a third on a six-slot wheel and it never rests; put a third on an eight-slot wheel and two pins meet in two slots and it jams. Both of those look like separate conditions and are one, and the boundary between them is an equation in integers with exactly three solutions.

 What one point can and cannot see of a group. For each of the twelve: the group's dimension, the dimension of one point's orbit under it, and the difference — the stabiliser, the motions that leave that particular point exactly where it is. The orbit is the only picture a group has, and this table is the honest caption on it. Planar motion and spherical motion are both three-dimensional and both sweep a point over a two-dimensional surface, so each leaves one motion doing nothing at all: a turn about the plane's normal in one case, a turn about the radius in the other. A point does not see the whole group, and no drawing of one trajectory can be a complete picture of what a joint permits. Drawn wrongly

Six things a joint is not

A freedom count read as a description, a screw system read as a group, a pair list read as a convention, a trajectory read as a determination, a nominal alignment read as a delivered one, and a higher pair read as a larger joint. Six claims, each of them what a careful person would say, each answered with a number.

 Whether the part goes in is one inequality. Two of the four contacts are moved and the other two left where the drawing says; the horizontal and vertical axes are those two errors, inward positive. The shaded region is where the part still goes in and the unshaded region is where it does not fit at all — not fits badly, not is located wrongly: there is no position and no orientation the part can take. The boundary is the straight line 0.250·e₁ + 0.250·e₂ = 0, whose coefficients are the shares from the previous figure. Four probe points are marked, each checked twice — once by the inequality and once by a linear program that looks for a pose and reports the program infeasible when there is none — and the two agree at every one. A hold turns a set of tolerances into a single condition, and the weights in it are what say which contact is worth making accurately. Contacts that only push

Which contact to make accurately

A hold turns a set of contact tolerances into one linear inequality, and the weights in it are the coefficients of the combination that cancels — a quarter each on a square held by four, and 0.144 to 0.424 on a hexagon held by five. Above that line the part goes in and below it there is no pose it can take at all: not badly located, not out of position, no fit.

 The machine compiled from a rectangular hyperbola. xy − 0.5, compiled: 20 bars and 20 joints, painted by what each part is for. The two-link arm at the pivot carries the tracing point; the reflectors and means build each term's angle; the rigid offsets fix the constant φₖ and the amplitude; the translators carry those directions out along the summing chain, whose last vertex is held on a line. That last constraint is the equation. Every joint drawn is the output of a Newton–Raphson solve on 36 equations, converged to 4.2e-16, and the polynomial at the tracing point is 2.2e-16. The curve as an equation

A compiled machine and its own scale

A linkage compiled from a polynomial has bars whose lengths are the polynomial's coefficients and joints whose angles are its phases. Scale it and every coefficient scales — so the machine computes the same polynomial multiplied by a constant, which is a different polynomial with the same roots.

 Two rotors cut undersize, and the gap that is their seal. The conjugate pair at three positions of its turn, each rotor cut 3 undersize — its boundary moved that far inward along its own normal, with the full-size outline dashed behind it. At full size the two are in contact at every angle, so cutting both back by 3 leaves exactly 6.0 between them wherever they were touching. Measured over a whole lobe pitch the gap runs from 5.9965 to 6.0000 against a prediction of 6, a worst departure of 3.46e-3. The shape is the unknown

The clearance that is the seal

Two rotors that are each other's conjugates touch at every angle of their turn, so cutting both back by the same amount leaves exactly twice it between them, everywhere. Open the shafts by the same amount instead and the gap runs from four per cent of it to all of it. And a pair that is not conjugate has no seal to cut: over one lobe pitch it swings from two and a half units inside itself to two and a third apart.

 A flat face on a swinging arm, and the cam it asks for. The cam a flat-faced follower needs when its face is carried on an arm pivoted 300 from the cam's centre, with a base circle of 20 and a lift of 20, drawn in the cam's own frame at three cam angles. The heavy line is the face, the dashed line is the perpendicular from the cam's centre to it, and the dot is the contact. Its distance from the foot of that perpendicular is the offset, which for a sliding face would be the programme's own velocity and here is a quantity with the arm in it. Over the turn it runs from -20.41 to 17.95, so the face must be 38.4 wide. Prescribed motion

A flat face on an arm is worse

Carry a cam's roller on a swinging arm instead of a slide and the pressure angle improves — a long arm beats an offset. Carry a flat face on one and the opposite happens. The cam sees the sine of the follower's rotation rather than the rotation, and the distortion costs convexity: the smallest workable base circle rises from 10.66 on a slide to 28.68 at a pivot three base circles out, and below a certain arm length no base circle works at all.

 The same five-bar with its gears meshed outside and inside. A geared five-bar drawn over a whole cycle at four ratios, with its two cranks turning against each other in the top row and together in the bottom one — an external mesh and an internal one, which is the same machine with the ratio's sign changed. Both assembly branches are drawn. The curves have the same degree in both rows — 1 : 1 at 6, 2 : 1 at 10, 3 : 2 at 16, 5 : 3 at 26 — and they are not the same curves: the lower ones are the maximally circular members of their degree and the upper ones are not, which is a difference nothing about the drawing shows. How many answers

The mesh inside keeps the half

Mesh a geared five-bar's two gears inside each other instead of side by side and the cranks turn together rather than against each other. The curve its pin draws has exactly the same degree at every rational ratio — and it passes through each circular point half that degree, which is as often as any curve can, where the counter-rotating machine manages between a third and two fifths. The sign of the ratio is the whole difference.

 A one-way clutch with no teeth in it. A bore of radius 20 with a 5-sided star inside it, each face a flat at 16.7 from the axis, and a roller of radius 1.6 in each wedge. The gap between a flat and the bore is widest at the middle of the flat and narrows towards its ends, so a roller rolled towards an end is nipped. Left: every roller at rest, 5.60° from its flat's own perpendicular, with 0.02 of clearance still in the gap. Right: every roller at its seat, 6.25° along, with the gap equal to the roller's diameter. Getting from one to the other takes 1.449° of the star, and that is the mechanism's lost motion. Motion that stops

A ratchet with no teeth

A roller clutch holds one way and runs free the other with nothing on it a tooth could be called. Whether it grips is a question about friction, which no drawing settles. How far it moves before it does is geometry: a clearance divided by the gap's own slope, so its lost motion is a length rather than a fraction of a pitch — 1.4° against a 24-tooth ratchet's 15° — and unlike a ratchet's it is not divided by adding more holding elements.

 One tooth count, three pressure angles. A 20-tooth gear cut at 14.5°, 20°, 25°, drawn at a common pitch radius and overlaid on its own pitch and base circles. The three teeth have the same thickness at the pitch circle — that is what the standard fixes — and differ everywhere else: a coarser pressure angle puts the base circle lower and leans the flank over, so the tooth gains metal where it is held and loses it where it reaches: thicknesses at the base circle run 1.628, 1.756, 1.967 modules and at the tip 0.866, 0.695, 0.510. Those two run opposite ways, and the second of them ends at a hard stop — a tooth whose tip thickness reaches zero has come to a point and cannot be cut. Teeth

The angle the standard left free

Involute geometry fixes the tooth curve and leaves one number open. Raising it buys smaller pinions and spends contact ratio, monotonically and in opposite directions, so there is no angle that is best at both — and the familiar twenty degrees is a choice with a date on it rather than an optimum.

 The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual. Links with a width

The gap is a straight line in the metal

Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

 The two curves the pole rolls along, at 60°. The pole is a different point at every instant, and it traces one curve in the fixed plane and another in the moving one. Those are the centrodes, and the whole motion is the second rolling without slipping on the first — a statement with no mechanism in it, which is why two completely different linkages with the same centrodes produce the same motion. The moving centrode is drawn here in the position it occupies at this instant, and it touches the fixed one at the pole to 0.0e+0 of a unit. positioned by solving, not by drawing. The motion, not the mechanism

The linkage, put back from two curves

This field opened by saying the mechanism drops out, and that every planar motion is one curve rolling on another. Both are true and neither had been measured. The rolling reproduces the four-bar's own placement to a residual that quarters when the sampling halves, and the two curves lay equal arc to a part in a billion.

 4 turns, and none of them in a plane. A strand wrapped 4 times round a drum of radius 50 whose barrel has room for 8, with each turn lying beside the last — so the pitch is the strand's own diameter, 10 here, and the path is a helix rather than a circle. The helix angle is 1.823°, the length per turn is 314.318 against the planar model's 314.159, and the whole run is 1257.27 long where a plane would have said 1256.64. The plan view is a planar wrap exactly: projected onto the plane perpendicular to the axis, the helix is a circle of the drum's own radius to 2.8e-14 traversed 4.000000 times. Members that pull

The wrap that walks along the axis

Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.

 Three joints, and the three screws that describe them. A leg of three revolute joints, drawn as its three axes, and the principal screws of the three-system they span, drawn through the system's own centre. The three principal axes are mutually perpendicular — worst cosine 1.1e-16 — and they meet at one point, missing it by 2.6e-16. Their pitches are -0.3766, -0.0338, 0.8075, and every screw the leg leaves free has a pitch the three of them give by h₁l² + h₂m² + h₃n². Nothing in the three joint axes looks like a right angle and the system's own frame is one. Out of the plane

What a leg of three joints leaves free

Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.

 Every way the same bars can be put together. The machine compiled from a rectangular hyperbola has 4 parallelograms, and a parallelogram's four bars also close as an antiparallelogram — so there are 16 ways to assemble it. One mark per way. 4 of them put the tracing point on the curve, 4 put it somewhere else, and 8 do not close at all. The ones that are wrong are not broken: they satisfy every bar to 9.6e-15 while the polynomial at their tracing point reads 1.5e-1. This is the gap in Kempe's original argument, and no tolerance on the closure could ever have found it. Drawn wrongly

Six things a compiled linkage is not

A closure residual read as a verdict, a theorem read as a design, an exact answer read as an accurate one, a degree read as a cost, a construction read as a search, and a neighbourhood read as a turn. Six claims, each of them what a careful person would say, each answered with a number.

 What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on. As built

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

 What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. The lit row is the one this essay is about. Of the 14 rows, 6 are quoted and 3 are exact. Machines you have met

Which numbers survived

Fourteen machines were measured against the numbers they are sold with. Three survived exactly, and all three are counts. One is false by a stated bound, two are exact means of things that vary, three are honest values at a stated position, and six name quantities their mechanisms do not have. The pattern is not about honesty; it is about what kind of thing a number is.

 The two circuits at a flat position, and the play that joins them. The input and output angles of a parallelogram whose input is short by 1e-3, near the flat position where the exact parallelogram's two motions would cross. With no play the linkage's configurations are two curves, an upper and a lower, 0.1095 radians apart in output angle at the flat input angle — the square root of the error, not the error. Each shaded band is what a radial play of a stated fraction of the error makes reachable, and the innermost boundary is the play-free pair. At a play equal to the error the bands meet and the linkage can pass from one circuit to the other. What can move

A length error is undone by its own size

A parallelogram built a thousandth wrong loses its change point, and the two motions it could have chosen between end up a tenth of a radian apart — the square root of the error rather than the error. The radial play that joins them again is a thousandth exactly: not of that order, that number. It is the same number a third crank charges the same linkage in misfit, and no pin is worth more of it than any other.

 Two cones, 20 teeth and 40, at 90°. The axial section, which is where every bevel quantity is read. Two cones share an apex and roll on one another along the element drawn heavy; their half-angles are 26.57° and 63.43°, adding to the shaft angle, and the ratio of their sines is the tooth-count ratio exactly. The dashed arc is the sphere of radius 22.36 on which a bevel tooth's profile actually lies. The two short lines perpendicular to the common element are the back cones; each is heading for its own axis at a distance r/cos δ from the pitch circle — 11.18 and 44.72 — and that distance is the pitch radius of the spur gear the tooth is really cut to. Both back cones lie on one line, because there is only one perpendicular to the pitch element at that point, and the two heavy stubs straddling it are the two teeth — each one addendum out from the pitch circle and 1.25 in. Teeth

A tooth that lives on a sphere

Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.

 Three circles at one point, at 50°. The osculating circle of the fixed centrode at the pole, the osculating circle of the moving centrode there, and the inflection circle. All three pass through the pole and all three have their centres on the common normal — which is what makes the relation between them a relation between three numbers on a line. The two centrode radii are 16.180 and 9.730, signed along that normal, and the inflection circle's diameter is 24.398 against the 24.409 the other two give. Nothing in that arithmetic knows there is a linkage. The window is 30.25 wide and the larger osculating circle is 32.36 across, so it is cut at the edge: what is drawn is the neighbourhood of the pole, where the three curvatures are the same three numbers however far the circles carrying them reach. The motion, not the mechanism

The two numbers are the curves' own

Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.

 What a leg of three joints permits and what it resists, on one frame. The leg of three revolute joints in grey, and the principal screws of two three-systems drawn through one centre: the twists the leg lets its platform make, and the wrenches it holds without any joint turning. They are computed separately — the second is the reciprocal complement of the first, found by its own eigenproblem — and they come out with the same centre, 5e-16 apart, and the same three axes, parallel to 0e+0. Only the pitches differ, and each wrench pitch is its twist's with the sign reversed, to 1e-15: -0.3766 against 0.3766, -0.0338 against 0.0338, 0.8075 against -0.8075. Out of the plane

The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.

 An inner cable in its sheath, pulled and pushed. A sheath routed through two bends — the first turning 90° on a radius of 70, the second turning back 90° on 55 — drawn with its bore exaggerated to a clearance of 7 so the inner's two positions can be seen. Pulled, the inner is taut and takes the shortest path the tube allows, hugging the inside of each bend; pushed, it is pressed against the outside. Both paths are strands over pulleys of radius R ∓ c at the bend centres. At this clearance the pulled inner is 20.53 shorter than the centreline and the pushed one 23.63 longer, against c times the total turning, 21.99. At a real clearance of 0.25 the pulled inner is short by 0.7834 against 0.7854. Dragging changes the first bend's angle. Members that pull

A strand in a tube

A Bowden cable's inner runs inside a sheath with a little clearance, and pulled it takes the shortest path the tube allows — a strand over pulleys of radius R − c at every bend. So its lost motion is the clearance times the total angle the sheath turns through: no bend radius in it, no route shape, and the S-bend that turns nowhere net loses as much as the U that turns back. Steering a handlebar changes it by exactly c times the change in angle.

 The same patch twisted 17.2°: 35 mechanisms, all at the edge. A rhombus of 8 × 8 kagome cells — 192 joints, 346 bars — with every up-pointing triangle turned by 17.2° about its own centre. Each joint is drawn with an area proportional to its weight: its share of the diagonal of the projector onto the patch's mechanisms, which does not depend on how the mechanisms are written down and adds up over the joints to the number of mechanisms, 35. That number is Maxwell's count, 2 × 192 − 346 − 3, and the rank agrees with no redundant bar. The mean weight is 0.341 on the outermost ring of cells and 0.015 on the innermost. The twist kinks every line of bars at every joint, and the mechanisms fall away from the edge 23-fold in 3 cells. Many of one thing

The count says how many and not where

A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.

 Six coupler points of one slider-crank: three at other distances from the crank pin, three exactly a rod away. A slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, drawn at one position with the closed curves six points of its rod trace over both assemblies. A point is given as (u, v) in the rod's own units — u along the rod from the crank pin, v across it. On the left (0.4, 0.5), (−0.3, 0.2), (1.2, −0.6): their distances from the crank pin are 0.64, 0.36, 1.34 rods. On the right (0.6, 0.8), (0, 1), (−0.8, 0.6), each exactly one rod from the crank pin, as far as the slider pin is. All six are quartics, all six are bounded — no real branch runs off the page — and nothing in the drawing tells the two panels apart. The difference is at infinity. How many answers

Where a slide puts the rest of the degree

A slider-crank's connecting rod draws a quartic that passes once through each circular point, which leaves two of its four meetings with the line at infinity unaccounted for. They are not along the slide. A point u along the rod and v across it sends them to the complex slopes [2v ± i(1 − u² − v²)] / [(1 + u)² + v²], whatever the crank, rod or offset — confirmed by slicing and by the fitted equation — and they are real only for points exactly a rod's length from the crank pin, where they merge into one direction at half the point's angle.

 The best four contacts on six regular polygons. The largest-margin placement of four frictionless contacts on regular polygons of 3, 4, 5, 6, 8, 12 sides, each found by exhaustive search over edge ends and refinement along the edges. 3 sides: margin 0.231 against a half-edge of 0.866; 4 sides: margin 0.333 against a half-edge of 0.707; 5 sides: margin 0.235 against a half-edge of 0.588; 6 sides: margin 0.293 against a half-edge of 0.500; 8 sides: margin 0.284 against a half-edge of 0.383; 12 sides: margin 0.223 against a half-edge of 0.259. From six sides up every contact sits at an end of its edge; on the triangle and the pentagon two of the four settle near the middles of edges instead. On the square and the even polygons the corner contacts take alternate ends of four edges a quarter-turn apart; an odd polygon has no edge exactly a quarter-turn round and holds less than either even neighbour. Contacts that only push

The hold is in the corners

A disc cannot be held by frictionless contacts and a regular polygon can, so a polygon with more and more sides has to lose its hold somewhere. Searched exhaustively, the best four contacts sit at alternate ends of four edges a quarter-turn apart, and their margin is the half-edge sin(π/n) less a correction that falls as 1/n² — 74% of it at eight sides, 99.4% at sixty-four. The hold is lost as the side shrinks, not as its square, and it is carried entirely by how far a contact sits from its edge's middle.

 One shaft angle, five places along a twisted rotor. A mismatched pair — a cycloidal rotor against a circular-tipped one of the same height, which are not each other's conjugates — with the rotors twisted by 1 lobe pitch along their length. The section at a given place along the shaft is the flat pair at an input angle shifted by the twist so far, so at one instant the five sections shown are at five different phases of the same mesh: the two bodies are into each other by 2.42 at the tightest section and 2.34 apart at the widest, with a mean of -0.408 across the whole seal. All five are drawn at one scale, and nothing here is meshed twice: one planar profile is read along a window. The shape is the unknown

A twist steadies what it cannot tighten

A helical rotor's sections are at different phases of the same mesh, so the clearance a machine has at one instant is a window along its own profile rather than a point on it. A wrap of exactly one lobe pitch holds the seal's open area constant through the turn — every harmonic at once, whatever the profile — and leaves its average, its tightest place and its widest place exactly where they were.

 A rectangular hyperbola, compiled from a multiple of its equation. The machine compiled from p · (1 + x² + y²) for a rectangular hyperbola, with the translators — the parallelograms that carry a direction from where it is produced to where it is needed — in their own colour. The factor 1 + x² + y² is at least one at every real point, so every equation here vanishes on exactly the same curve. The machines do not agree: 20 bars at p, 75 bars at p · (1 + x² + y²), 144 bars at p · (1 + x² + y²)². This one solves 29 positions over an arc of 0.508 radians, and at every one of them the original polynomial reads 4.93e-14. The curve as an equation

The price is on the equation

A line costs five bars. The same line, written as its own equation multiplied by a factor that is never zero, costs fifty — and the machine compiled from the longer equation draws the same line just as exactly. Every cost this field quotes belongs to a polynomial and not to a curve, and the cheapest equation of a given curve is a quantity nobody here has.

 What a sweep at the wrong resolution reports. One machine — a crank passing a stud a tenth of a unit across — swept at six sample counts. At eight samples the smallest gap found is 0.0809 and the machine reads as clear; at twelve it is 0.0067 and still clear; from sixteen on it is negative and the crank is inside the stud by 0.0099. Nothing about the twelve-sample answer looks wrong: the curve it draws is smooth, its minimum is interior, its margin is small and positive. A sweep cannot report what it did not look at, and the repair is not more samples but a bound on what happens between them. Drawn wrongly

Six things a body is not

A verdict read as a measurement, a hull read as a part, a sweep read as a proof, a drawing read as a configuration, a geometry read as a force, and a plane read as a place. Six claims, each of them what a careful person would say, each answered with a number.

 Three machines, one curve. The coupler curve of a four-bar, drawn three times by three different linkages. Roberts's theorem gives every coupler curve exactly three four-bars that trace it, and their proportions are not close: the cranks here are 1.600, 2.214, 2.558, a spread of 60%. Read as an identification problem this is a least-squares objective with three separate exact minima and nothing between them — so an instrument that records only where the tracing point went has three answers however good it is, and no amount of data chooses. What chooses is knowing where the ground pivots are, which is a different measurement rather than a better one. Numbers that were measured

Three machines, one curve

Roberts's theorem says every four-bar coupler curve is drawn by exactly three different four-bars. Read as an identification problem that is a least-squares objective with three separate exact minima, whose cranks differ by sixty per cent — so an instrument that records only where the tracing point went has three answers, and no amount of data chooses between them.

 A framework built off square, and the play it asks for to move at all. The nine bars with their two lines 4° from perpendicular, as built and after the first joint has been pushed 0.2 along its line. The framework has a freedom by rank and no motion, so the push cannot be taken with the bars at their lengths. With every bar allowed to be wrong by 1.509e-4 — which is a radial clearance of 7.547e-5 at each end — a placement exists, and the worst bar in it is out by 1.509e-4. The same push on the perpendicular framework needs no allowance at all, because there it is a motion. What can move

The right angle as a tolerance

Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then the framework reaches its first crossing and the law is left three hundred times behind.

 24 teeth inside 72. An internal pair, drawn from the same involute the external pairs are drawn from. Three things are different and they are one difference. The centre distance is 24.0 — the difference of the pitch radii rather than their sum. Both base tangency points lie on the same side of the line of action, so the pitch point falls outside the segment between them rather than inside it. And the contact ratio is 1.931 against the external pair's 1.707, because the annulus's tip circle cuts the line on the far side of its own tangency and lengthens the contact path instead of shortening it. The pinion turns the same way as the annulus, which no external pair ever does. Teeth

The mesh with one curvature reversed

Turn an annulus's teeth inward and the same involute law produces a different machine: a centre distance that is a difference, two base tangencies on one side of the line of action, more contact than an external pair carries, and three separate floors on the tooth counts, all of them the same statement about where an involute stops existing.

 Holding the coupler still: the frame's inflection circle is the mirror of the coupler's. The four-bar with ground 4, crank 1, coupler 3.5 and rocker 3 at a crank angle of 225°, redrawn in the coupler's own frame, so the coupler is the horizontal bar and the ground link is what moves. The circle on one side of the pole is the ordinary inflection circle — coupler points travelling straight — carried into this frame. The circle on the other side is the inverse motion's: frame points travelling straight when the coupler is held. It is computed by differentiating the swap and, separately, by solving the coupler-held four-bar, and both routes put it at the first circle reflected through the pole, to 7e-15 on a radius of 4.45. The line through the pole is the common tangent. Dragging the crank angle moves both circles and they stay mirror images. The motion, not the mechanism

The frame seen from the coupler

Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.

 Two circles, one term. The curve r² = 1.44 together with r² = 2.56, whose squared radii sum to 4.00 — four times the square of the arm's link length. Their product equation expands to 1 term, which is what either circle costs on its own, so the second circle is free. The machine compiled from it has 17 bars against 11 for the single circle, runs over 5.200 radians against 1.560, and stays on the outer component throughout: its radius varies by 4.15e-13 over 240 solved positions. A mechanism moves continuously and the two circles are disjoint, so no assembly of it reaches both. The curve as an equation

Two circles for the price of one

Search every multiple of a curve's equation by a polynomial of degree two and the cheapest is the curve's own equation, on four curves and by exhaustion. On the circle a second multiplier ties — and what it describes is two concentric circles, whose squared radii sum to four times the arm's link length squared, at exactly the cost of one.

 The taut inner, and the walls it is actually held by. A sheath of two 8° bends on a radius of 40, its bore drawn at a clearance of 4, with the shortest path from ferrule to ferrule inside it. Nothing about a bend goes into that path: it is the shortest route the tube allows, found by tightening a funnel against every cross-section in turn, and the places it reaches the wall are where it is held. At this clearance the two bends hold 53% and 53% of their own turn, and the inner is short of the centreline by 0.854 against the closed form 1.117 — 76% of it. Widen the bore and the path lifts off. Members that pull

Which walls a strand is held by

A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.

 What each branch carries, against what the engine delivers. The power in the variator's branch and in the straight path, as multiples of the engine's own, for a planetary of K = 1.40. Both are ratios of powers, so the load cancels and nothing here is a force: the split comes from requiring the gearset to be lossless at every admissible set of speeds, which fixes the torques at 1 : K : −(1+K). The variator carries v/(K − v) and the straight path K/(K − v), and both run away at the pole. The variator first carries the engine's whole power at v = 0.700, which is exactly half the way to the pole — and the straight path is already carrying more than the engine everywhere past nought, flowing the other way through the planetary. That excess is the circulation. More than one input

The power that goes round twice

Sliding a power split's travel towards its pole buys ratio span for nothing, on the kinematics. It is not for nothing. The variator's own branch carries v/(K−v) of the engine's power and overtakes it at exactly half the way to the pole, and the tolerance trim the span was computed from is not reached until the variator is rated for six times the engine — which no machine is.

 Where a tilted platform is still singular. A slice of the workspace at height 2.4, at the dead yaw and 3° of tilt, with each position shaded by how well the platform is held there — dark where the six legs are nearly dependent and pale where they are not. Level, this whole square would be uniformly dark. Tilted, the dark places are a curve through it, which is what an ordinary direct singularity looks like on a slice. Driving a search downhill from forty-eight starts reaches a smallest singular value of 0.00e+0, so the locus is a singularity rather than a shallow valley. Several legs, one platform

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

 A four-bar at 1°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 3.3e-14 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 54.3°. Drawn wrongly

Six things a measurement cannot tell you

A calibration with a perfect residual whose fourth number is a starting guess, a rank that says nothing about a second answer, an improvement that proves nothing about a parameter, a class with no margin, a plan scored on poses that were refused, and a model missing something no data can find. Six claims, each with the number that kills it.

 The field of action, straight and slanted. The rectangle a contact lives in: its width is the usable line of action, its height is the face width, and a new contact line enters every base pitch. On the left the teeth are straight, so the lines are vertical and each one arrives and leaves all at once — the total length in contact jumps by a whole face width, 40 mm, every base pitch. On the right the same pair with a 20° helix: the lines lean by the base helix angle, so a tooth enters at one end of the face before it has left at the other, and the total changes by 3.44 mm instead of 40. Nothing about the profile is different between the two panels. Teeth

Contact that runs along the tooth

A straight tooth engages along its whole face at once, so the amount of contact at a mesh is a square wave. Slant the tooth and a second contact ratio appears that has no tooth count in it and no pressure angle — bought with face width and helix angle alone, and able to reach a value at which the total length of contact stops varying at all.

 Three laws in the band: a third, a sixth and a sixth. For bands from 3% to 10 ppm of the output's swing, three things measured on the symmetric six-bar itself: the longest dwell at any angle at the rocker pin, the dwell at the sixth-order angle γ = 120.8024°, and how far below γ the best angle sits. The dots are measurements and the lines are the two-term model's predictions from two coefficients read off the output's even part. Over the four tightest bands the fitted exponents are 0.173, 0.172 and 0.333, against a sixth, a sixth and a third. At 10 ppm the best angle is 0.490° below γ against a prediction of 0.488°, and the dwell at γ is 26.43° against 26.23°. The loosest bands sit above the lines, where the terms the model leaves out are no longer small. The paths points trace

The dip that buys the dwell

A symmetric coupler curve's six-bar dwells longest a little below the angle that makes its error sixth order, and three bands gave three numbers. Across three and a half decades of tolerance the best angle's distance from that angle grows as the cube root of the band, both dwells as its sixth root, and their ratio is (27/4)^(1/6) = 1.3747 at every band — because the best machine spends the whole band on one dip and ends its dwell where the output comes back.

 The same multiplier, applied to the error. A tong whose units are cut to an angle 0.01 radians away from the drawing. If one unit is out, the span is out by that unit's share and nothing more; if every unit is out the same way — which is what a machine setting or a worn tool produces — the error is multiplied by the unit count, exactly, to 7.6e-14. The third column is what would happen if the errors were independent and equally likely either way: the accumulation goes as the square root of the count instead, and the difference between the two columns at thirty-two units is a factor of 5.66. Which column applies is a question about how the parts were made, not about the mechanism. As built

The error that is repeated

Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.

 Two circuits at an ordinary change point, and how far apart a length error leaves them. A four-bar on the Grashof boundary g + a = b + c — ground 4, crank 1, coupler 2.50, output 2.50 — with its crank short by 1e-3, near the one input angle at which its two assemblies would meet. Built exactly, the two curves would cross at the origin. Built with the error they pass each other 0.0400 radians apart at the flat input angle, against the law 2√(2bδ/c(g + a)) = 0.0400, and the pin clearance that rejoins them is 1.0000e-3: the error itself. Dragging the coupler's share of b + c moves the separation and leaves the clearance where it is. What can move

Every change point lies flat

A parallelogram a thousandth wrong is rejoined by a pin clearance of exactly a thousandth, at any of its four bearings. The obvious guess is that an ordinary change point — a linkage on one Grashof boundary with no equal bars — would need a clearance with a constant in front and would reveal which bearing is loose. It does neither, because every change point has its four joints on one line. What does acquire a constant is the angle: 2√(2bδ/c(g + a)) when the circuits separate, and a stall constant with no coupler or output in it at all.

 The same links and pins, and between 12 and 15 link lengths in the stack-up. Every closed loop in a mechanism is one equation a tolerance analysis has to satisfy, and the equation involves every link the loop passes through. All 16 chains here have the same number of independent loops — 3, which is pins minus links plus one and is fixed by the two totals — but not the same shortest set of them. The bars are the total length of a minimum cycle basis, and they run from 12 to 15. So the smallest number of link dimensions that any stack-up on this mechanism can involve is decided by the graph, before a single dimension has been chosen, and two topologies a count cannot tell apart differ by 3 of them. As built

Where the shortest loops are

A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.

 Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance. As built

The pair a catalogue sells

A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.

 One bar made 0.0001 too long, one bar at a time. Every bar of the machine compiled from a lemniscate lengthened by 0.0001 in turn, the machine re-solved, and the polynomial read at the tracing point. It is no longer zero anywhere. The worst bar takes it to 9.1e-3 — an amplification of 91 — and the median bar to 2.8e-4. The bars that matter are the reflectors, which are the cheapest part of the machine; the translators, which are most of it, barely move the answer at all. Size and fragility live in different parts. As built

Exactness a micron destroys

Lengthen one bar of a compiled machine by a ten-thousandth and its tracing point leaves the curve. On the smallest machine the error comes out smaller than it went in; on a fifty-bar one it comes out ninety times larger — and the bars that matter are the reflectors, which are the cheapest part of the machine.

 The same part, dimensioned two ways. Three holes at 0.00, 1.40, 3.50, each located to ±0.010 by the process. Dimensioned as a chain, each hole is placed from the last, so the two gaps are ±0.010 each and the errors accumulate over the whole span to ±0.0141. Dimensioned from a baseline, every hole is placed from H0, so the span is ±0.010 and the gap between H1 and H2 — which touches the datum at neither end — is ±0.0141. The two schemes are exactly √2 apart in opposite places. Nothing about the part decides this and nothing about the process does; the drawing does, and a tolerance analysis that starts from the lengths has already thrown the information away. Numbers that were measured

The same part, dimensioned twice

Three holes, one process, two drawings. Dimensioned as a chain the errors accumulate and the span is ±0.0141; dimensioned from a baseline the span is ±0.0100 and the gap between the second and third holes is ±0.0141. Exactly √2 apart, in opposite places, and nothing about the part or the process decides which — the drawing does.

 A clearance that has a sign, over a tolerance box. The same machine at each of the sixteen corners of a ±0.02 band on its four lengths, with the smallest gap over a whole drive computed at each. The nominal machine clears by 0.1041; the worst corner clears by 0.0763 and the best by 0.1321, a band 0.0558 wide from a tolerance of 0.02 on each length. Every other quantity computed from the lengths becomes an interval when the lengths do. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together. 24 random draws inside the box beat no corner, which is the check that the extremes are where they are assumed to be. As built

A clearance inside a tolerance box

Every quantity derived from the lengths becomes an interval when the lengths become ranges. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together.

 A number that runs away. The Denavit–Hartenberg offset of a pair of nominally parallel axes, against how far from parallel they actually are, for a fixed out-of-plane tilt of 0.05°. The marks are extracted from the geometry by finding the common normal; the line is the closed form −A cos α cos β sin β / (sin²α cos²β + sin²β), and the two agree to 2.4e-16 relative over three decades. At 30° of twist the offset is 0.0030 of a link length; at 0.01° it is 1102. The dashed line is the worst case over the tilt, which sits at β = α and is exactly A/2α. Nothing about the machine has changed by as much as a degree. Numbers that were measured

A number that runs away

A hundredth of a degree of unintended twist on a nominally parallel pair of joint axes puts the Denavit–Hartenberg offset at −1,102 link lengths. The extraction from the geometry and the closed form agree to 5 × 10⁻¹⁶ over three decades, and the worst case over the tilt is exactly A/2α.

 The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer. As built

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

 Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number. Numbers that were measured

Which numbers have a size

Take every length in a mechanism up and down together and fit the power each computed quantity follows. A transmission angle lands on zero, a coupler point's speed on one, a path curvature on minus one, an enclosed area on two — and a tolerance band held to a fixed ±0.01 lands on minus one, which nobody would guess.

 What each instrument recovers. The same twenty poses of the same four-bar, read three ways. A protractor on the output link recovers 3 of the four lengths and leaves the fourth exactly invisible, because its readings are dimensionless in the lengths and scaling the machine does not move them. A coordinate machine on the tracing point recovers all six parameters — the four lengths and the two that say where the tracer sits — at a condition number of 162.3. Using both recovers the same six at 26.1, 6.2 times better, which is the case for putting two instruments on one machine: not more parameters, better-conditioned ones. Numbers that were measured

What this field cannot measure

Every field on this site has a boundary and this one has three: a direction the readings cannot span, an alternative no derivative detects, and a model nobody thought of. The first is computable exactly, the second needs a search, and the third is not detectable from data by any method at all.

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