Which numbers survived

The field opened with fourteen machines and fourteen numbers and a promise to check them. This is the check.

The count #

Six of fourteen name a quantity the mechanism does not have. That is the field’s headline and it is worth being careful about what it does and does not say.

It does not say the numbers are wrong. A wrong number is one whose value differs from the true one, and for six of these rows there is no true one to differ from. A roll centre height is a construction that returns a different answer at every position; “100% Ackermann” is a percentage of a condition met at two angles; a scissor lift’s rise per unit of ram is a ratio of two quantities that are not proportional. Being wrong is not available to any of them.

It also does not say anybody was careless. Every one of the six is a convention — a value at a design position, an idealisation, a name for a family of behaviours — and inside the trade that uses it, everybody knows which. The failure is what happens when the number leaves the trade.

The three that survived, and what they have in common #

A scissor lift has one degree of freedom. Not approximately one. Mobility is the nullity of a constraint matrix and a nullity is an integer.

A kinematic coupling is exactly constrained. Six independent contact wrenches, rank six, nothing free and nothing doubled. Also a rank.

A compound epicyclic reduces 20.528301886…:1. Exactly 2176/106, because the reduction is a ratio of tooth counts, and a tooth count is a count.

All three are counts. That is not a coincidence and it is the field’s most transferable finding.

A count cannot vary with the mechanism’s position, because it has no position-dependence to vary with. It cannot be quoted at an operating point, because it has no operating points. It cannot be a mean, because it does not fluctuate. It is the one kind of quantity whose exactness is structural rather than approximate, and every exact number in fourteen machines is one.

The three that are honest without being exact #

Three rows are neither exact nor quoted, and they are the ones worth imitating.

A chain drive’s 53/11 is a mean, and the mean is provably the tooth ratio — it follows from both sprockets passing the same pins, and the measurement confirms it to five figures without being told. A quoted mean is a good specification as long as the thing it is a mean of is stated.

A Watt linkage’s “it locates the axle” is bounded. The claim is false by 34 microns over eighty millimetres of travel, and the error is fifth order in the travel — so the reader can compute their own number for their own suspension. That is the most useful row in the table.

A rocker ratio is an operating point and the trade knows which point: zero lift. It is honest because the convention is established and because there is an established practice of measuring the thing the convention is about.

The pattern is that each of them attaches something to the number: a period to average over, an order and a bound, a position. The rows that fail are the ones where the attachment was dropped.

The ledger as an instrument #

It is worth saying how the table works, because the table is the field’s only piece of machinery that is not a mechanism.

Each row is a small object naming a machine, the number it is sold with, the verdict, the library function that produced the measurement, and a note. The measured column is a string rather than a number on purpose: the whole point of a row is that the quoted and measured quantities cannot be subtracted, and giving them a common numeric type would invite exactly that.

The rows are computed once per build and memoised, so the fourteen measurements — five suspension sweeps, a steering scan, two axle sweeps, a scissor sweep, a lock sweep, two chain runs, a valve sweep, two coupling analyses and a door swing — cost about half a second in total and are shared by every essay that draws the table.

And the ledger checks itself in two ways. It refuses a row whose verdict is not one of the five it defines, which is the failure that would otherwise be silent: a typo in a verdict makes a row vanish from every count that groups by it and the table still renders. And it asserts that at least one row exists for each verdict, which is what keeps the closing arithmetic above from being about a table that has quietly lost a category.

Those are small checks and they are the kind this site keeps finding it needs. A figure that draws is not a figure that is right, and a table of measurements is exactly as trustworthy as its ability to notice that one of them has gone missing.

What a good specification would look like #

Nothing in this field argues for more decimal places. What it argues for is a change of shape, and there are four available.

A range, with the sweep it was taken over. “Roll centre 73 mm static; 52 to 106 over ±80 mm of travel” is three numbers instead of one, and every suspension package computes all three already.

An order. “Lateral movement under 0.05 mm at 80 mm of travel, growing as travel⁵.” A reader can rescale it.

A stated position. “Rocker ratio 1.605 at zero lift; 1.588 at peak.” Two numbers, and the second is the one that decides the valve lift.

A count, where the design can be made to turn on one. This is the strongest and the least often available: a quantity that is an integer is a quantity nothing can erode.

The first three cost a line of output from a computation that has already been done. That is the practical conclusion of the field, and it is not a demand for rigour — it is an observation that the information exists and is being discarded at the last step.

What the field measured that was not a number at all #

Two findings from the phase are not about specifications, and they are the ones most likely to be useful elsewhere.

A roller is not a slider, and the difference is one equation. A scissor lift counted with prismatic pairs comes out as a structure with mobility −1, at every stack height, for a machine holding a car in the air. The count was correct arithmetic about a wrongly described mechanism — which is a much commoner failure than the famous case where the formula cannot see the geometry.

A hidden pin cannot open a door. The corner of a door moves sideways at a rate equal to how far the pin sits behind it, so with two doors touching there is no pin position behind the door’s own face that clears. That is a theorem with a one-line proof and a map that confirms it, and it is why a concealed hinge is a four-bar: a centre can be where a part cannot.

What the field got wrong on the way #

Four mistakes are recorded in the essays, and they are recorded because the pattern in them is the useful part.

The steering error came out at 55° because the outer wheel’s angle was mirrored along with the arms. Fifty-five degrees is the scale that says sign rather than linkage.

The Watt axle linkage wandered 39 mm — eleven times worse than the Panhard rod it is supposed to beat — because it was built in a configuration that is a perfectly good five-bar and not a straight-line mechanism. What caught it was the fitted order: 0.88, which is not a power law any linkage produces.

The chain drive reported 0.02% fluctuation for two identical sprockets in step, which must be exactly zero. It was the Runge–Kutta integrator’s, and replacing the integration with the closed form the geometry actually has removed it.

The coupling’s freedom came out as a rotation about an axis it does not have, because a routine returning a list of eigenvectors was read as returning columns. The magnitude was right; the direction was invented. It was caught because three parallel grooves let a part slide, and sliding is not turning.

All four produced plausible output. Three of them were caught by a quantity whose value was known in advance — a sign, an order, a physical direction — and the fourth by a case with a known answer. Nothing was caught by inspection of the numbers.

What the field did not do #

The four-bar hinge. The map says one must exist; the synthesis machinery is here; a hinge that fits inside a cupboard was not produced. Making one is a constrained synthesis — reach three poses, land the pivots inside the carcase, clear everything in between — rather than a construction, and it is the first thing a phase after this one should build.

The rack and pinion. Most cars steer through one, it has one more design parameter than a trapezoid, and that parameter is exactly what lets it do better. Two closures instead of one, and no new machinery.

Bump steer, castor and steering axis inclination. All three are couplings between the suspension and the steering, all three are spatial, and every mechanism in this field is planar.

The cycloidal drive. The other way to make a large reduction from a difference of integers, with a drawable geometry and an equally exact ratio.

The scissor lift as a spatial mechanism. Two assemblies in parallel planes joined by cross-shafts, which is overconstrained in exactly the way the spatial field’s loops are, and counted with the six-freedom arithmetic rather than the three-freedom one.

Anything with a force in it. Which is the same boundary as ever, and it is worth restating in this field’s terms: a latch’s security is computed here as a distance, not as a load; a chain’s fluctuation as a velocity ratio, not as a tension; a coupling’s sensitivity as a displacement per unit of contact error, not as a stiffness. Every one of those has a force answer that follows from the geometry by virtual work, and every one of them is somebody else’s essay.

Where the field sits in the site #

Twelve fields now, and this one is a lens rather than a subject. Every essay in it deepens a ladder one of the first six fields opened: three on mobility, four on the four-bar, two on the coupler, two on teeth, two on cams, and these two on the things drawn confidently and wrongly. Not one of them needed kinematics the site had not already built.

What it needed was two small additions and one absence made good. A prismatic pair between two moving links — because a scissor lift’s top roller and a strut’s rod both slide along something that is itself moving, and the solver had only a ground-fixed slide. And bodies with a width and a signed clearance between them, because a door has to miss its neighbour and a link with no width cannot. And the absence made good is the one what-is-still-outside named a phase ago: interference between links, listed as the site’s most obvious geometric omission, now answered for one mechanism and left open for the rest.

That is the shape a field like this should have. The machines are the reader’s; the questions are the site’s; and the new machinery is whatever the machines forced.

Making a specification turn on a count, and what it costs #

Prefer a count is the strongest of the four recommendations and the one described above as least often available, and it is worth pressing on that, because it is available more often than it looks and the price is quotable.

The move is always the same shape: replace a continuous quantity with a discrete one by adding teeth, detents or contacts. A friction drive’s ratio is a length and varies; a toothed belt’s is a count and does not. An adjustable stop is a position and drifts; a detent plate is an integer and does not. A part held by clamping force is located by whatever the force does; a part on six contacts is located by geometry. In each case the design gains an exact specification and gains it for the reason the three surviving rows survived — the quantity has stopped depending on where anything is.

The price is resolution, and this site has measured it twice. A compound epicyclic’s reductions near twenty to one are spaced forty per cent apart, so exactly twenty is not on offer. A reverted train needs a sixty-three-tooth wheel for twelve to one and fifty-six for sixteen. A ratchet’s resolution is its pitch, and no accuracy improves it. Exactness is bought by giving up the ability to have any value at all, and the values on offer are decided by the arithmetic rather than by the designer.

So the recommendation, stated properly, is a trade rather than an improvement: a count gives an exact value from a sparse set, and a length gives any value inexactly. Which is preferable depends entirely on whether the requirement has latitude — and that is the same question the Diophantine essays in this collection keep arriving at from the other side.

That also explains why the three exact rows are the machines they are. A scissor lift’s freedom count and a coupling’s exact constraint are counts nobody chose to make discrete; they are discrete because degrees of freedom are integers. Only the compound epicyclic is a case where a designer converted a ratio into a count deliberately, and it is precisely the one that pays the sparseness — its available reductions are widely spaced and it is quoted at 20.528301886 rather than at a round number, which is the price written on the specification for anybody to read.

Which gives the field’s advice its honest final form. Ask what varies; attach a period, a bound or a position to any number that does; and where a count is available, take it and quote the sparseness alongside it. The last clause is the one this field would add to the usual advice, because a specification that is exact and unattainable at the wanted value has solved the wrong problem.

The thing worth carrying #

The field’s method was three questions, and they cost nothing.

What varies? Not “is this right” but “does this quantity depend on the mechanism’s position, and if so, which position is this?” Six rows fail here before any arithmetic.

What is it a mean, a bound or a value of? A number with something attached — a period, an order, a position — can be used by somebody who does not share the conventions it was written under. A bare number cannot.

Is there a count available? Where a design can be made to turn on an integer, the resulting number is exact and stays exact, and every exact number in this field is one.

Those three are the applied lens, and each of them is cheap. The first can be asked of a drawing. The second can be asked of a specification. The third can be asked before a design exists, and it is the only one of the three that changes what gets built.

The reason the field is arranged by ladder rather than by machine is that they are not about machines. A scissor lift and a kinematic coupling have nothing in common; the questions asked of them are the same questions the constraint field has been asking since what decides whether it moves, pointed at a specification sheet.

There is one more thing the ledger says that no single row does. Of the eight rows that are not exact, seven describe mechanisms that work perfectly well — the suspension steers, the chain drives, the latch holds, the axle stays put — and the failure is entirely in the description. Only one row describes a mechanism that does something its user would not want: the door that passes through its neighbour. So the field is not a catalogue of bad mechanisms. It is a catalogue of good mechanisms described in a format that cannot hold what they do, and the format is the thing worth changing.