Every rational gear ratio has a degree

One freedom, and a motion that never repeats put a gear on each crank of a five-bar and asked whether the machine ever comes home. At a ratio of 3 to 2 it is back after two input turns; at 37 to 23 after twenty-three; at the golden ratio never. Its closing section asked the next question about the same machine. When the motion does close, the pin where the two couplers meet draws a closed curve, and that curve is algebraic, the zero set of some polynomial. What degree does the polynomial have, and how does it grow with the ratio?

That essay proposed answering it with fits of rising degree. A fit is one of three instruments used here, and the weakest. The answer comes from counting, twice over, and it turns out to be a formula: at ratio p : q in lowest terms the pin’s curve has degree 4·max(p, q) + 2·min(p, q).

One turn of a smaller angle #

The machine is the one the closing essay built: two cranks of length 1 on pivots 3 apart, couplers of 3 and 2.5 meeting at a pin, and gears that make the second crank turn k times as fast as the first, the other way. With k = p/q in lowest terms, the input turns q times while the output turns p times, and then the machine is home.

That suggests the variable to write it in. Let the input angle be qφ and the output angle be a fixed offset less pφ. As φ goes once round from nought to a full turn the input makes q turns and the output p, so one turn of φ is exactly one cycle, and every configuration of the cycle is reached once. The crank pins’ coordinates are cosines and sines of qφ and pφ, which are polynomials in cos φ and sin φ of degrees q and p. The pin is fixed by two circle conditions about the crank pins. Everything is a polynomial in four unknowns, cos φ, sin φ and the pin’s two coordinates, with the identity $\cos^2\varphi + \sin^2\varphi = 1$ tying the first two together.

So the curve is algebraic at every rational ratio, and the question of its degree is well posed. At an irrational ratio there is no such variable, since no angle turns a whole number of times while both cranks do, and the motion never closes.

Both of the pin’s positions are drawn in the figure, one per colour. For each pair of crank angles the two couplers can meet on either side of the line joining the crank pins, and a polynomial cannot choose between the two, so the curve whose degree is being asked is both branches together, exactly as a four-bar’s coupler sextic is both assemblies’ ovals.

A tracker that stops at five to three #

The first instrument is the one the curve nobody eliminates used on a six-bar: cut the curve with a random complex line and count where the pin can be. The four polynomial equations and the line condition are handed to homotopy continuation, which tracks one path for each solution of a start system with the same degrees.

The degrees are worth keeping small. Written naively, the circle about the first crank pin has degree 2q, because the pin’s squared distance from its pivot comes out as a polynomial of that degree even though it is always 1. On the unit circle it is 1, so it can be replaced by 1 before the system is written, and likewise for the second crank. That leaves equations of degree 2, q + 1, p + 1 and 1, and a Bézout number of 2(p + 1)(q + 1): 8 paths at 1 : 1, 24 at 3 : 2, 252 at 13 : 8.

At 1 : 1 the tracker’s 8 paths give 6 points. At 2 : 1, 12 paths give 10, and at 3 : 2, 24 give 16; every path either arrives or leaves for infinity, and none stalls. From 5 : 3 on it does not finish. Of 48 paths, 22 stall, and the arrivals give 24 points. At 8 : 5, 67 of 108 stall and the count is 31; at 13 : 8, 181 of 252 stall and the count is 40.

A stalled path is one whose step size shrank below its floor before it reached the end, and a count made with stalled paths is a lower bound, since every missing path could have been a meeting. The tracker has run out of reach, and the paths that leave explained the likely reason: surplus paths heading for solutions at infinity of high multiplicity approach them slowly and look, to a step-size rule, exactly like paths in trouble. From 5 : 3 on, the tracker alone cannot say what the degree is.

One polynomial in one variable #

The second instrument removes the tracking altogether, and it comes from the coordinates nine times through each circular point used to explain circularity.

Write positions as Z = x + iy and Z̄ = x − iy, treated as independent, and write the angle as $w = e^{i\varphi}$. The first crank pin is then $r_1 w^{q}$ and its partner coordinate is $r_1 w^{-q}$; the second crank pin is the pivot’s position plus a constant times $w^{-p}$, and its partner has $w^{p}$. Every pin coordinate is a Laurent monomial in the single variable w, a power that may be negative.

The pin is on both circles. Each circle condition is a product of a Z-difference and a Z̄-difference, and subtracting one from the other cancels the pin’s own ZZ̄ and leaves an equation linear in Z and Z̄. A line is another linear equation, aZ + bZ̄ + c = 0. Two linear equations fix Z and Z̄ by Cramer’s rule, each as a ratio of Laurent polynomials in w, and putting them back into the first circle condition leaves one Laurent polynomial F(w) whose roots are the configurations with the pin on the line.

A polynomial in one variable has exactly as many roots as its degree, and they can be found all at once. The Aberth iteration takes every root of F simultaneously, with no deflation. Each root is then checked against the machine rather than against F: the pin it gives is put back into both circle conditions and the line, and the worst relative residual over every root at every ratio is 3.9 × 10⁻¹³. Every root is a configuration, and at a random line every configuration gives a distinct point.

The roots agree with the tracker at 1 : 1, 2 : 1 and 3 : 2, where it finished, and go on where it could not: 26 at 5 : 3, 42 at 8 : 5, 68 at 13 : 8, 110 at 21 : 13 and 178 at 34 : 21. The largest is a polynomial of degree 178 in one variable, and every one of its roots gives a pin on the line.

The span is the degree #

A Laurent polynomial that runs from $w^{-m}$ to $w^{n}$ is $w^{-m}$ times an ordinary polynomial of degree m + n, so it has m + n nonzero roots. The degree of the curve is therefore the span of F’s exponents, and the span can be read off the pieces F is built from without solving anything.

At 3 : 2, the first crank pin is $w^{2}$ and its partner $w^{-2}$; the second is a constant plus $w^{-3}$ and its partner a constant plus $w^{3}$. The determinant of the two linear equations mixes both and runs from $w^{-3}$ to $w^{3}$. The first factor of F, the pin’s Z less the crank pin’s, multiplied through by the determinant, runs from $w^{-3}$ to $w^{5}$, and the second from $w^{-5}$ to $w^{3}$. Their product runs from $w^{-8}$ to $w^{8}$, while the determinant’s square only reaches $w^{-6}$ to $w^{6}$ and cannot cancel the ends. F spans sixteen.

The same bookkeeping in general, with q the larger of the two: the first factor reaches $w^{2q}$ through the crank pin times the determinant and $w^{-p}$ through the second pin, the second factor mirrors it, and the product spans 2(2q + p). That is 4·max(p, q) + 2·min(p, q), and at every one of the eight ratios the roots found number exactly that. The argument needs the two extreme coefficients not to vanish, which a special machine could arrange and a random line does not; the eight counts are the check that this machine is not special.

The formula makes the growth along the golden ratio’s convergents explicit. Each convergent’s numerator and denominator are consecutive Fibonacci numbers, so each degree is four times one Fibonacci number plus twice the one before, and the sequence 6, 10, 16, 26, 42, 68, 110, 178 is itself a Fibonacci sequence. Each degree is the sum of the two before it, and the ratio of successive terms is 1.618 by 110 to 178. The motion that comes nearest to closing at those turns does so on curves whose degree grows without bound.

The real roots are the drawing #

The roots of F live in the complex w-plane, and the drawing lives on its unit circle, since a real angle φ is a w of modulus one. A real line across the drawn curve should therefore meet it at exactly the roots that lie on that circle.

The line cos 0.3 · x + sin 0.3 · y = 1.8 crosses the 3 : 2 curve at eight places, four on each branch. F for that line has sixteen roots, the degree, and eight of them lie on the unit circle to seven decimal places. The other eight are scattered inside and outside it, at complex angles, and are the meetings no drawing shows.

The same line was checked at five ratios. At 1 : 1 it crosses twice on each branch and F has four roots on the circle; at 2 : 1, four and two crossings and six roots; at 3 : 2, eight and eight; at 5 : 3, six and eight crossings and fourteen roots; at 8 : 5, ten and ten and twenty. Every count of roots on the circle equals the crossings of both branches together, found separately by stepping along the traced curve and watching for a change of side. The polynomial’s roots and the machine’s drawing are the same set, and neither branch alone accounts for them.

What a fit can see, and at the golden ratio #

The third instrument is the one the closing essay proposed: trace the curve, fit polynomials of rising degree, and see where a null space opens. A null space of fifteen is not noise set out what such a fit reports and when to believe it, and one of its findings matters here: one piece of a curve decides far less than the whole. So both branches are traced.

At 1 : 1, 1,442 points on both branches give nothing at degree five, whose largest drop is 2.8; one curve at degree six behind a drop of 4.9 × 10¹²; and three at degree seven behind 5.0 × 10¹¹, the sextic times the three polynomials of degree one. At 2 : 1 the fit finds nothing at nine, one at ten behind 1.0 × 10¹⁰, and three at eleven. At 3 : 2, with 2,882 points, degree fourteen gives nothing and degree sixteen gives one curve behind 3.9 × 10⁶.

The drops fall as the degree rises, twelve decades at six, ten at ten, six at sixteen, and that is the fit approaching its reach rather than the curve changing character. At 5 : 3 the fit would need 378 coefficients, and the figure does not attempt it. The Stephenson arm curve was out of a fit’s reach at degree eighteen, with every real configuration traced; this curve is still inside it at sixteen, and two degrees and two different curves are not enough to say where the reach ends or what sets it.

At the golden ratio the fit is given 3,602 points over twenty input turns, both branches. At degree six its largest drop is 2.0, at ten it is 2.6, and at fourteen it is 3.4. No polynomial of degree fourteen or less comes anywhere near vanishing on the pin’s path. That is what the counting predicts: there is no finite degree to find, since the convergents’ degrees already pass fourteen at 3 : 2, and the pin’s path over many turns fills an area, as the closing essay measured, rather than lying on any curve.

A gear passes the circular points less often than a pin #

The circularity essay found that every curve drawn by a machine made only of pins passes through each circular point half its degree times, and gave the reason: every bar is a Z-difference times a Z̄-difference, and nothing in such a machine mixes the two otherwise. A slide broke the rule by tying a pin’s Z to its own Z̄. A gear ties something too, and the same instrument, a line through a circular point, measures what.

A line through a circular point is aZ + c = 0 or bZ̄ + c = 0, and the polynomial route handles it with one extra care. With b nought, F acquires a factor of the second crank pin’s coordinate less the first’s, and its p + q roots are the angles at which the two crank pins’ Z-coordinates coincide, where subtracting the circles decides nothing and no pin is given. Those roots fail the check against the machine and are set aside: exactly p + q of them at every ratio, through either point.

The genuine roots then number 2(p + q) through each circular point: 4 at 1 : 1, 6 at 2 : 1, 8 at 3 : 1, 10 at 3 : 2, 14 at 5 : 2, 16 at 5 : 3 and 26 at 8 : 5. The tracker, run on the same lines at the first four ratios, gives 4, 6, 8 and 10, with three and six paths stalled at 3 : 1 and 3 : 2 and the counts unaffected. So the circularity is the degree less 2(p + q), which is 2·max(p, q): 2 at 1 : 1 where half the degree would be 3, and 16 at 8 : 5 where half would be 21. The curve falls short of half its degree at each circular point by min(p, q).

The reason is visible in the Laurent monomials. The first crank pin’s Z is a positive power of w, and the second crank pin’s Z is a negative power. As w grows, the first pin’s Z runs off while its Z̄ shrinks, which is the pin behaviour, but the second pin does the opposite at the same time: its Z̄ grows and its Z settles. The mesh has tied the Z-side of one crank to the Z̄-side of the other, which is exactly the separation the pin argument depended on. Some of the curve’s meetings with the line at infinity are then pulled away from the circular points, and min(p, q) of them at each.

Eight ratios, one machine, external gears #

The degree formula is argued from the exponents and checked at eight ratios on one machine. The argument assumes F’s two extreme coefficients are nonzero. For a machine with special lengths, or a gear offset that puts the crank pins in a special phase, they could vanish and the degree drop; no such machine is surveyed.

The circularity 2·max(p, q) is measured, not derived. Seven ratios agree with it and the reason given is a direction, not a count. Why the shortfall is exactly min(p, q) is not shown.

Only external gears. The cranks here turn in opposite senses. An internal gear, or a belt, turns them the same way, which changes the signs of the exponents on the second crank pin, and neither the degree nor the circularity of that machine is measured.

The tracker’s stalls are not diagnosed. They are consistent with slow paths to solutions at infinity, and a projective formulation or an endgame at the end of each path might reach 13 : 8. That would be a better tracker, not a different answer: the roots of F already give the count.

What comes next: the same gears turning the same way #

The same gears turning the same way. With an internal mesh the second crank’s angle is the offset plus pφ, its pin’s Z carries $w^{p}$ and its partner $w^{-p}$, so both crank pins’ Z-coordinates would grow together as w does, as a pin machine’s do. The span argument gives that machine’s degree from its pieces in a few lines, and the circularity instrument would say whether a co-rotating gear keeps the half that a counter-rotating one loses.

How many double points the geared curve spends at infinity. A curve of degree 16 passing six times through each circular point uses thirty of its double points there. The configuration curve here is a double cover of a circle, since each φ has two branches, and its genus is set by where the two branches meet. Counting those branch points, as never three circuits did for the six-bars, would give the genus and so the number of finite double points the 3 : 2 curve must have, a number the drawing can partly check by its visible crossings.

The fit’s reach, measured on one family. The drops at the true degree were twelve, ten and six decades at 6, 10 and 16. A family whose degree can be dialled in steps of one would show the reach of a double-precision fit as a single number, the degree at which the drop stops reaching rounding, and whether that number depends on the curve or only on the count of coefficients.