Never three circuits

The component count settled what a four-bar’s branches are. They are the connected components of its configuration space, a crank-rocker has two, a linkage that fails Grashof’s condition has one, and Grashof’s inequality decides which.

What it did not settle is whether two is as many as there can be. Every four-bar drawn so far came out at one or two, and a collection of examples that never shows three is not a demonstration that three cannot happen. The question is worth asking properly, because the answer is not “a four-bar has two assemblies at each crank angle, so it has two circuits”. That sentence sounds like a reason and is not one, and a six-bar will show why.

This essay measures the bound first and then finds out what sets it.

What is being counted #

A circuit is a connected component of the configuration space: a set of configurations the machine can move between without being taken apart. For a four-bar the space is a curve on the torus of two angles — crank and rocker — and a circuit is one closed piece of that curve.

The curve in that figure was found by contouring the closure condition on a grid and linking the pieces, and its count is a measurement at the grid’s resolution. A census of many thousand linkages wants something better than a grid, and a four-bar allows it.

A count with no grid in it #

Put the crank pin at angle θ. Its distance from the far ground pivot is fixed by the law of cosines, and the far dyad — coupler and rocker — can reach across that distance only if it is no longer than their sum and no shorter than their difference. So the machine assembles at θ exactly when cos θ lies in a band between two numbers computed from the four lengths.

A band on a cosine can cut the circle of crank angles in only four ways. It can contain the whole circle, or one arc centred on the ground line, or one arc centred on the far side of it, or two arcs placed symmetrically, or nothing.

Over an arc the two assemblies of the dyad exist inside the arc and merge at both ends, where the dyad goes straight, so the arc carries one closed circuit. Over the whole circle they never merge, so there are two circuits, each going all the way round. That is an exact count for any four lengths, with no sampling in it anywhere.

It was run on twenty thousand four-bars, the ground held at one — a circuit count does not change with scale — and the other three lengths drawn uniformly between 0.05 and 3. That range is wide enough that all four Grashof classes appear, along with a great many linkages that cannot be assembled at all.

Every one of the 3,031 crank-rockers has two circuits. So does every one of the 3,086 double cranks and the 1,497 Grashof double rockers. Of the 12,386 linkages that fail Grashof’s condition, 9,528 have one circuit and the other 2,858 cannot be assembled. Not one linkage in twenty thousand has three, and not one that fails Grashof’s condition has two.

The census is shown at the head of this essay. It also closes a loose end: the exact count agrees with the contour on every linkage the contour has been run on.

Where the assemblies merge #

The ends of the arcs are where the count comes from, so they deserve a name and a picture. At each end the two assemblies of the dyad coincide. In the vocabulary of a curve covering the circle of crank angles twice, each end is a branch point. In the vocabulary of a machine, it is a limit position: the crank can go no further and the linkage turns back onto its other assembly.

Read the three rings together and a rule appears. Where there are real branch points, the circuits are the arcs between them, two branch points to an arc. Where there are none, the linkage either assembles everywhere, with two circuits, or nowhere, with none. So the count of circuits is half the number of real branch points, or two when there are none.

The double rocker is the case that shows the rule is not a restatement of crank and rocker. Its crank does not turn fully, and it still has two circuits: two separate arcs of crank angle, each about forty degrees wide, each carrying its own closed loop.

The count has a visible counterpart in the plane. A coupler point carries each circuit to a closed curve, and on the curves field’s census of 160 four-bars every Grashof linkage drew two ovals and every other linkage one, with no two circuits of one linkage drawing the same oval. So the bound proved here for circuits is also a bound on the number of separate pieces a four-bar’s coupler curve can have, and it is two for the same reason.

That is the same number the crank-rocker has, reached by a different shape. A property that survives a change of shape like that is usually a property of something underneath the shape, and here it is the branch points.

Why four, counted in the complex numbers #

A branch point is where the crank pin is exactly the dyad’s full length from the far pivot, or exactly the difference of its two lengths. Each is a pair of circle conditions: the crank pin on its circle about the first pivot, and on a circle of the stated radius about the second. Two circles meet in two points in the complex plane, whatever their sizes — Bézout’s four, less the two circular points at infinity that every pair of circles shares.

So a four-bar has four branch points in the complex numbers, always: two where the dyad is straight and two where it is folded. The continuation tracker counts them rather than taking that on trust. It follows all four Bézout paths of each pair of circle conditions, and for the site’s crank-rocker two paths arrive and two leave in each pair, four distinct points in all.

What varies from one linkage to the next is how many of the four are real, and a real circle pair meets in two real points or none. The real branch points therefore come in pairs: none, two or four. That is the whole of the reason for the bound. Four branch points at most can be real, a circuit that ends needs two of them, and so no more than two circuits can end — and a linkage with no real branch points has its two circuits and no more.

The count is exactly four only when the four points are distinct. At a change point two of them collide: for a four-bar with ground, crank, coupler and rocker all equal there are two distinct branch points, and for one with lengths 4, 1, 3 and 2 there are three. That collision is the self-crossing of the configuration curve the component essay described at the parallelogram’s flat position, seen from the other side.

From four branch points to a genus #

The rule above is elementary, and it is also the shadow of a theorem that does not stop at four-bars.

Complexify the crank angle and the circle of crank angles becomes a sphere. A four-bar’s configuration curve, over the same field, covers that sphere twice — two assemblies over every point — except at the four branch points, where the two sheets join. The Riemann–Hurwitz formula says what surface a cover like that is. For a double cover of a sphere with four simple branch points, twice the genus less two equals twice the sphere’s two less two, plus four: the genus is one.

A four-bar’s complex configuration curve is a torus. The branch-point count is not a mnemonic for that; it is what the genus is computed from.

The last step is Harnack’s theorem, from 1876, which is cited here rather than proved: a smooth real algebraic curve of genus g has at most g + 1 real components. For genus one that is two.

So there are now two reasons a four-bar never has three circuits. One is the band on a cosine, which is short, specific to the four-bar, and complete. The other is genus and Harnack, which is longer and says nothing about four-bars in particular. The second is worth having only if it keeps working where the first stops, and the next machine up is where the first stops.

Six-bars, where the band argument gives out #

A six-bar is a four-bar with a second dyad hung from one of its moving bodies. Watt’s chain hangs it from the rocker, Stephenson’s from the coupler, and both have mobility one. At a given crank angle each has up to four assemblies — two for the four-bar, and two for the second dyad over each of those.

The band argument does not transfer. The second dyad closes when its near pin is at the right range of distances from its own ground pivot, and that pin is not on a circle about the crank’s centre. As the four-bar goes round a circuit, the pin’s distance from the new pivot can rise and fall several times, and every entry into and exit from the band is another pair of branch points.

The genus does transfer, and the tracker can count it. A six-bar’s configuration curve is a double cover of its four-bar’s curve, branched where the second dyad goes straight or folds. Those are the four-bar’s loop closure plus one more circle condition, and tracking all sixteen Bézout paths finds four points of each kind for the site’s Watt six-bar and six of each for its Stephenson six-bar. Covering a torus twice with eight branch points gives genus five, and with twelve gives genus seven. Harnack then allows six circuits and eight.

The two counts differ because of where the pin rides. On Watt’s rocker it moves on a circle, which a circle about the new pivot meets twice, and each rocker position comes from two four-bar configurations. On Stephenson’s coupler it moves on a coupler curve, a tricircular sextic, which a circle meets in only six finite points, because six of its twelve meetings are used up at the circular points.

The census says the bound of two is a fact about four-bars. Of three thousand Watt six-bars, 527 have four circuits. Of three thousand Stephenson six-bars, 373 have four, nine have five and two have six. Neither chain comes near its genus bound: none of the Watt six-bars has five, and none of the Stephenson six-bars has seven.

The refuted sentence at the top of this essay is now visibly wrong. A Stephenson six-bar has at most four assemblies at any crank angle, which is no more than a Watt six-bar has, and it can have six circuits. The number of assemblies over a point does not bound the number of pieces.

Four and six, drawn #

A census of random machines finds the common counts and rarely the extreme ones, so a search was run for each chain: ninety restarts of a random climb that keeps any change which does not lose a circuit. It counted each candidate at two resolutions and kept the smaller count, so that a circuit thinner than the sampling could not be counted. The best of each is recorded, and it keeps its count at every resolution from 240 to 8,000 samples a circuit.

The Stephenson six-bar’s four-bar is a Grashof double rocker with a margin of two hundredths — its coupler is the shortest link and nearly too long to be — so it has two circuits, each an arc of crank angle. Along each, the coupler point enters and leaves the second dyad’s band of distances three times, and the six-bar has six circuits where its four-bar has two.

The Watt example is the more striking of the two, because its four-bar fails Grashof’s condition and has a single circuit. The machine hung beneath that one circuit has four.

Why the search stops two short #

Both chains stop two below Harnack’s number, and the branch points say what reaching it would take.

A six-bar circuit that ends does so at two real branch points, so the circuits that end number at most half the real branch points: four for Watt’s chain, six for Stephenson’s. The only circuits that need no branch points are those over a whole four-bar circuit along which the second dyad closes the entire way round, and each of those gives two. Harnack’s six and eight are exactly half the branch points plus two. Reaching them needs every branch point real, lying on one of the four-bar’s circuits, while the dyad closes all the way round the other.

The recorded machines have the first half and not the second, and the search found no machine with both. Whether one exists is not settled here, and it may be that the geometry of a coupler point forbids it. The census and the search are lower bounds on the true maximum, and Harnack’s number is an upper bound on it. Neither is claimed to be the maximum.

For the four-bar the two bounds meet at two: a census that reaches two and a theorem that allows no more. That meeting is what makes never three a statement rather than a tally.

What the count rests on #

Four things, stated so that the two parts of the result are not read as equally strong.

The four-bar bound is a theorem. The band argument proves it for every four lengths, and Riemann–Hurwitz with Harnack proves it again. The census confirms both and adds nothing to their certainty; what it adds is the distribution, and the fact that every Grashof linkage attains two.

The genus is counted, not quoted. The branch points come from tracking every Bézout path and counting distinct arrivals to one part in a hundred thousand. Harnack’s theorem itself is cited.

The six-bar counts are measurements at a resolution. A circuit narrower than the sampling would be missed, which is why the recorded machines are re-counted at five resolutions and why the census’s own maxima are reported as what 300 samples a circuit finds.

Smoothness is assumed where it is generic. Harnack’s bound is for smooth curves, and every machine measured here has distinct branch points. A special six-bar whose branch points collide would have a singular configuration curve and a different count. The change-point four-bar is the example of that happening, and it is exactly where the component count had already found a curve crossing itself.

What comes next: the genus of a curve somebody can draw #

The genus in this essay belongs to a configuration curve, which no machine draws. It has a visible counterpart.

A point on a six-bar’s second dyad traces a curve in the plane, and when each point of that curve is drawn by exactly one configuration, the traced curve is the configuration curve in another coordinate system, with the same genus. The Stephenson six-bar’s arm point is such a curve. Its genus is the seven counted above, and its degree has never been fitted, because fitting it would need more coefficients than the arithmetic can separate.

A degree counted on a line built the instrument that does not need a fit. The next essay turns it on the Stephenson six-bar’s arm, to find the degree of a curve whose equation nobody has written down: the curve nobody eliminates. A further question is open here as well. Whether a six-bar can reach its Harnack number is a question about arranging real branch points, and it is the natural continuation of this one.