One freedom, and a motion that never repeats

This field began with one question about a mechanism — whether it moves — and has answered it by two instruments ever since. Grübler’s count reads the numbers of links and joints and never looks at a length. The rank of the constraint Jacobian reads the lengths and never asks what a joint is. When they agree, the number they agree on is called the mobility, and a mechanism with a mobility of one is described as having one freedom: turn its input and everything else is decided.

A wheel forced the first split in that word, between what a mechanism can do at an instant and what it can reach over an interval. This essay makes a second split, on a mechanism where the two instruments agree completely and there is no velocity constraint anywhere. One freedom, both routes, no disagreement, and a question about the motion that the number does not answer: whether it ever repeats.

A five-bar with its cranks geared together #

Put two cranks on fixed pivots three units apart, each one unit long, and join their pins by two coupler bars, of 3 and 2.5, that meet at a third pin C. That chain is a five-bar, and it has two freedoms: both cranks can be turned independently, which is why a drawing robot built on it needs two motors.

Now fix a gear to each crank and mesh them. The second crank is no longer free. If the gears’ pitch radii are in the ratio k, the second crank turns k times as fast as the first and in the opposite direction, because an external mesh counter-rotates. For a ratio of 3 to 2 on centres three units apart, the pitch circles have radii 1.8 and 1.2, and they touch at a point on the line between the pivots.

The lengths were chosen for a reason that matters to everything below. However the two cranks sit, their pins are between 1 and 5 units apart, and the two coupler bars together can span anything from 0.5 to 5.5. So the chain closes at every pair of crank angles, with room, and there is no position anywhere at which it jams, folds flat or has to choose between ways forward. Whatever happens to this machine’s motion is decided by the gear ratio and by nothing else.

The count and the rank #

Grübler’s count for the geared chain takes five links — the frame, two cranks with their gears, and two coupler bars — five pins, and one more joint: the contact of the teeth. A tooth contact is a higher pair, touching along a line that travels over both flanks, and it removes one freedom rather than two. So the count is 3(5 − 1) − 2·5 − 1, which is one.

The rank sees the same thing differently. The three moving pins have six coordinates between them. There are four equations holding the four bar lengths, and a fifth holding the mesh: the second crank’s angle plus k times the first crank’s angle is a constant. The mesh equation’s derivative with respect to each pin points along that pin’s direction of travel round its pivot, and the matrix of all five derivatives has rank five. Six coordinates less a rank of five leaves one.

Without the mesh, both give two. With a mesh, both give one, at every ratio tried: 1 to 1, 2 to 1, 3 to 2, 5 to 3, 37 to 23, the golden ratio and the square root of two. The rank is not a close decision at any of them — the smallest singular value of the Jacobian, over four input angles, is 0.418 at its lowest and 0.642 at its highest, nowhere near the scale at which a rank could be misread.

Notice what the ratio does to that matrix. It appears in one row as a coefficient multiplying the first crank’s direction of travel, and any non-zero value leaves the row independent of the other four. So the instruments are not merely agreeing at each ratio; they are structurally unable to tell one ratio from another. Whatever distinguishes a 3-to-2 machine from a golden-ratio one, it is not a mobility.

What the ratio decides #

Something does distinguish them, and it is easiest to see by drawing the configuration space. With no gears, a configuration of this chain is a pair of crank angles, and since every pair closes, the configuration space is every pair. An angle and an angle plus 360° are the same angle, so the space is a square whose left edge is glued to its right and whose bottom is glued to its top — a torus.

The mesh draws a line on that torus. As the first crank turns through α, the second turns through −kα, so the configuration moves along a straight line of slope −k. When it leaves the square through one edge, it re-enters at the same height through the opposite one.

After N turns of the input, the second crank has turned kN times, and the machine is back in its starting configuration exactly when kN is a whole number. For a ratio of 3 to 2 that first happens after 2 turns, with the second crank having made three; the line on the torus has crossed the square in six strands and closed. For a ratio p to q in lowest terms it first happens after q turns, and never at a fraction of that: before q turns, the second crank’s angle has only ever missed its start by some multiple of a q-th of a revolution.

For an irrational ratio it never happens at all. No whole number of turns multiplied by an irrational number is a whole number, so the line on the torus never meets its own start. After twelve turns at the golden ratio it has crossed the square in thirty-three strands and ends 149.9° short of home.

This is not a new fact about lines on a torus. What is worth having is where it lands. Mobility is a statement about the neighbourhood of a single configuration: how many directions the linkage can move in from here. Whether the motion repeats is a statement about the whole of one orbit, and the orbit can be a closed loop or a line that never closes while every neighbourhood along it looks exactly the same.

A ratio of tooth counts is always rational #

Real gears settle the question for most machines before it is asked. A toothed gear pair’s ratio is a ratio of two tooth counts, which is a ratio of two integers, so every geared five-bar built from toothed wheels is a rational machine. It comes home, and the only question is when.

The answer can be long. A pair of 37 and 23 teeth has a ratio of 1.6087, within six-tenths of a per cent of the golden ratio, and since 37 and 23 share no factor, the machine comes home after 23 turns of the input and not before.

Over those 23 turns, pin C traces a curve that crosses itself many times and fills a band of the plane densely enough that the eye cannot find its end. It is nevertheless a closed curve of finite length, and the twenty-fourth turn retraces the first exactly. The same thing is true of a clock train, whose ratios are chosen from integers for exactly this reason: what an integer ratio buys is a mechanism whose state after a known number of turns is known.

An irrational ratio needs a transmission without teeth. A friction drive, a belt on smooth pulleys or a variator can in principle be set to any ratio at all. In practice a ratio like that is a measured quantity with slip and a tolerance on it, and asking whether it is exactly rational has no physical answer. The irrational machine below is therefore the model’s statement, an idealisation made to see what the ratio is doing, and it should not be read as a machine anybody has built.

The golden ratio never comes home #

Trace pin C for twenty turns of a golden-ratio mesh and the curve is still open: after twenty turns the pin is 2.056 units from where it started, which is as far away as it could reasonably be.

That single number is misleading in the opposite direction from a closure, because some turn counts bring the machine much nearer home than others. The right way to see which is to plot, turn by turn, how far the second crank’s angle is from its starting value.

The distance falls, jumps back up, falls again, and every so often sets a new record. The turns at which it does are 1, 2, 3, 5, 8, 13, 21, 34, 55 and 89. They are the Fibonacci numbers, and the reason is the continued fraction.

Every irrational number has a sequence of best rational approximations, its convergents, read off its continued fraction, and no fraction with a smaller denominator comes closer. The golden ratio’s continued fraction is the simplest there is, a one followed by ones for ever, and its convergents are ratios of consecutive Fibonacci numbers: 2/1, 3/2, 5/3, 8/5, and so on. A record return after N turns is a whole number M for which the golden ratio times N is closer to M than it has been for any smaller N — which is the same thing as M/N being a best approximation. So the records are the convergents’ denominators, and the measurement finds exactly those and no others.

It also finds how close each gets. At 89 turns the second crank is 1.8090° from home, and 89 times that gap, as a fraction of a whole turn, is 0.4472. That number is 1/√5, and it is not a coincidence of this machine. It is Hurwitz’s theorem, which is quoted here rather than proved: every irrational number has infinitely many approximations p/q closer than 1/(√5 q²), and no larger constant than √5 works for all of them. The golden ratio is the number for which that bound is tight. Of all irrational ratios, it is the one that keeps the machine farthest from home for longest.

The same measurement at the square root of two gives records at 1, 2, 5, 12, 29 and 70 turns, the denominators of its convergents, and 70 turns times the gap settles at 0.3535, which is 1/(2√2). Different number, different constant, same structure.

Home, and nearly home #

The 37-to-23 mesh is close to the golden ratio, and on the same turn-by-turn plot it looks like it for a while.

For twenty-two turns the gaps behave irregularly, and not one of them is small: the nearest the machine comes to home before the twenty-third turn is about a twenty-third of a revolution, as the arithmetic of a ratio with denominator 23 requires. At the twenty-third turn the gap is exactly nought, and from there the pattern repeats with period 23.

The contrast with the golden ratio is the whole of the essay’s distinction, drawn on one kind of plot. An irrational mesh approaches home ever more closely and never arrives. A rational one stays a fixed distance away and then arrives exactly. Closeness to the golden ratio does not change that. A 34-to-21 pair, a convergent, is closer still, within a tenth of a per cent where 37 to 23 is within six-tenths, and it too is periodic, coming home after 21 turns. Every toothed pair is periodic however close it comes, and the golden-ratio machine they approximate is not periodic at all.

How much of the plane the pin covers #

There is a third way to see the difference, and it is the one that shows where the distinction stops being visible.

Lay a grid of small cells over the plane and count the ones pin C has passed through. At 3 to 2 the pin covers 0.96 square units in two turns and never covers any more, because every later turn retraces those two. The golden-ratio pin keeps adding area — 3.32 square units after ten turns, 4.40 after twenty — until by fifty turns it has covered 4.86, and after a hundred and two hundred 4.87. It stops growing there not because the motion has closed but because it has visited every cell of the region pin C can reach at all. A line that never closes on a torus passes arbitrarily close to every point of it, and the pin’s region is the image of the whole torus.

The 37-to-23 pin covers 4.42 square units after twenty turns and 4.55 after fifty, then stops, because it closed at twenty-three. At this grid’s resolution it has covered all but a sliver of what the irrational pin covers. That is the honest limit of the distinction. Periodic and non-periodic are different in kind, exactly, in the model, and at any finite resolution a rational ratio with a large enough denominator is indistinguishable from an irrational one. The number of turns before a rational machine repeats can be made longer than anything anybody will run it for.

What mobility is a statement about #

Two instruments measure mobility, and this essay is not a case where either is wrong. Both are statements about an instant: how many independent directions of motion are available at the configuration the linkage is in. For this chain the answer is one everywhere, at every ratio, and the answer is correct.

What neither can see is the orbit. A one-freedom mechanism’s motion is a curve through its configuration space, and whether that curve closes is a property of the whole curve rather than of any piece of it. The two can differ arbitrarily: a closed loop of length two turns, a closed loop of length twenty-three, and a line that never closes, all with identical neighbourhoods everywhere along them.

The pairs field has met the same phenomenon in a different guise. A screw whose pitch is irrational generates a one-parameter family of displacements whose closure is larger than the family, for exactly the reason the golden-ratio line fills its torus. And the field’s census of freedoms as groups is a classification of what a mechanism can do near the identity, which is also local. So it is not that the instruments were blind to something they should have seen. They answer a local question, and periodicity is a global one.

What the model leaves out #

Backlash and tooth errors. A real mesh has play, so the second crank’s angle is not exactly k times the first’s but somewhere in a band round it, and the exact closure at 23 turns is a closure to within the backlash, accumulating nothing because the teeth re-engage each time.

Slip. A friction drive’s ratio drifts with load and wear, so the idealised irrational machine is at best a machine whose ratio is not known well enough to say which kind it is.

Where the pin goes. The coverage figure measures cells visited and says nothing about how evenly or how often. The fraction of time the golden-ratio pin spends in each part of its region is a further measurement and is not made here.

What comes next: a curve with no equation #

The next question is about the curve pin C draws rather than whether it closes, and it lands in the curves field’s question of what equation a coupler curve satisfies. A four-bar’s coupler curve is an algebraic curve of degree six. A geared five-bar’s coupler curve at a rational ratio is also algebraic, because the crank angles are tied by a polynomial relation among their cosines and sines, but its degree grows with the numerator and denominator of the ratio. At an irrational ratio the curve is not the zero set of any polynomial at all.

Its distinct argument would be a measurement of that growth: fitting polynomials of rising degree to the traced curves at 1 to 1, 2 to 1, 3 to 2 and 5 to 3, finding the degree at which each fit’s singular-value gap opens, and showing that a one-freedom mechanism built from pins, bars and two gears can draw a curve that no degree of fit will ever capture.