A machine with one dyad in it

The topology field’s question is which parts of a mechanism have to be solved together. Its answer is Assur decomposition: peel off groups of links that become rigid when their outer joints are held, and what is left is what cannot be peeled.

A four-bar has nothing to peel. Hold its two ground pivots and the remaining two joints are determined together, not one after the other — two circles meeting in two points is a dyad, and a dyad is the smallest thing that must be solved as a pair.

That is at four links. This rung applies the same walk to a machine of four hundred.

The walk #

Start with the grounded joints. Then repeatedly: place any joint whose position is fixed by things already placed.

A joint is fixed when it has two independent conditions against placed parts. Two bars to placed joints — two circles, two intersections. A rail whose line runs between two placed joints, plus one bar. A slide, plus one bar. Or a rigid attachment to a placed link, which is two conditions on its own and is why a rigid offset has one placement where a pair of bars has two.

When nothing can be placed alone, look for a dyad: two unplaced joints joined by a bar, each with one other placed anchor. Place both at once, and continue.

If the walk finishes, the machine has no simultaneous solve larger than a dyad in it. If it stalls, what remains is an Assur group of higher order, and the topology field’s census is about which of those exist.

What it finds #

The hyperbola’s machine: twenty joints, sixteen placed alone, one dyad.

The cubic’s: sixty-nine joints, sixty-five placed alone, one dyad.

The quintic’s: two hundred and forty-two joints, two hundred and thirty-eight placed alone, one dyad.

The dyad is the same one every time, and it is the arm. Drive the crank and the arm’s first joint is fixed; its tip and the fourth vertex of the parallelogram carrying the second arm angle back to the pivot are then determined together, because each has one bar to a placed joint and one bar to the other.

Everything else — every reflector, every mean, every rigid offset, every one of the hundred and fifty-four translators — places alone.

Why it comes out that way #

Because the machine was built that way, and the building is what the gadget architecture is.

Every gadget places its output from parts that already exist. A mean’s far vertex is fixed by two bars to links already at the pivot. A reflector’s output is fixed by a bar to its own far vertex and a bar to the pivot, and its far vertex is fixed by a bar and the mirror line. A translator’s output is fixed by two bars to joints one hop back along the chain.

So the arithmetic half of a compiled machine is a directed acyclic graph: nothing is ever built from something built from it, because the multiples strictly decrease at every step of the recursion, and the summing chain runs outward from the pivot and never back.

An acyclic construction unwinds. That is not a discovery about compiled machines; it is what construction means, and the walk is the check that the implementation actually has that property rather than merely being described as though it did.

The walk had to be taught about dyads #

The first version of the walk placed joints one at a time and nothing else, and it stalled immediately on every machine — with every free joint unplaced, including the arm’s.

That is the right answer to the wrong question. An undriven compiled machine has two degrees of freedom, so nothing in it is determined at all, and a walk looking for determined joints correctly finds none. Driving the crank fixes one, and the arm’s tip and the parallelogram’s fourth vertex then form a pair each of whose members has one placed anchor and one bar to the other.

Adding the dyad step is what makes the walk answer the question the field actually asks — what has to be solved together — rather than what is determined by two bars. Without it the walk reports every mechanism in the subject as completely undetermined, which is true and useless.

A four-bar is the case that forces this, and it is the reason Assur’s decomposition starts at dyads rather than at links. Any walk that cannot place a dyad cannot place a four-bar, and a tool that cannot handle the smallest interesting mechanism in the subject is not a tool.

What makes this worth measuring #

Two things, and the second is the one that belongs to this field rather than to the computing one.

The check catches a class of error. A compiler that accidentally built a loop — two gadgets each taking the other’s output — would produce a machine that still solved, because Newton does not care about construction order, and whose behaviour would be subtly different. The walk would stall on it. Nothing else in this field’s machinery would notice.

The result separates two axes that get conflated. A mechanism can be large and it can be deep, and this site’s own vocabulary has not kept them apart. The topology field’s interesting objects are deep: chains with Assur groups of order two and three, where several joints have to be found at once and no construction order exists. A compiled machine is enormous and has none of that.

Four links can be deeper than four hundred. The four-bar is the smallest mechanism in the subject with an irreducible pair, and the quintic’s compiled machine has exactly the same irreducible content — one dyad — in sixty times the parts.

Where a compiled machine sits among the field’s own objects #

The topology field has a scale of its own, running from chains that fall apart into dyads to chains that do not, and it is worth locating a compiled machine on it precisely.

A slider-crank is a dyad on a driven crank: shallow, four links, and every position solvable in closed form. That is why it appears in every textbook’s first chapter.

A four-bar is one dyad, and it is already past what a construction handles without a choice: two circles, two intersections, and which one is an assembly branch rather than a computation.

A Stephenson six-bar contains an Assur group of order two, and its solution needs a resultant. The field’s dwell mechanism is one, and its position analysis is genuinely harder than a four-bar’s rather than merely longer.

A compiled quintic is at the shallow end with the slider-crank, four hundred bars along.

That ordering is the point. The field’s difficulty scale is not the field’s size scale, and a reader who has spent a phase on censuses of ten-link chains might reasonably assume the two are related. They are not related at all, and a machine sixty times larger than the largest object in the census is at the easy end of the scale the census is about.

What the depth would have cost #

It is worth saying what would be different if the walk had stalled.

A machine with an Assur group of order two contains four joints that must be found together, which means a polynomial system with a resultant and more than one solution to choose between. At order three it is worse. Every such group is a place where the mechanism’s configuration is not determined by construction and where a solver may find a different answer than intended.

A compiled machine has one such place — the arm — and its two solutions are the arm’s two elbow positions, which is the ordinary two-branch situation of any two-link chain.

That does not mean a compiled machine has only two assemblies. It has $2^k$, and the branch census counts them. The difference is where they come from: not from irreducible groups, but from each individually-placed joint having two circle intersections to choose between. A machine can be shallow and still have an enormous number of assemblies, and the two facts are about different things — depth is about what must be solved together, and branch count is about how many answers each solve has.

Counting the two kinds of placement #

The walk reports two numbers per machine and the split between them says what the machine is made of.

On the quintic: two hundred and thirty-eight placed alone, one dyad. Of the two hundred and thirty-eight, most are translator outputs — one joint each, two bars each, a hundred and fifty-four of them — and the rest are reflector and mean vertices and rigid offsets.

The rigid offsets are the interesting minority. An offset’s joint is placed by a single constraint that supplies two conditions, so it has one placement rather than two, and it contributes nothing to the branch count. Every other joint in the machine contributes a factor of two.

So the walk incidentally counts something the branch census needs: how many of a machine’s joints have two placements. On the quintic that is two hundred and forty-two joints less the offsets, which is why the census caps what it enumerates and reports a lower bound rather than a total.

The order the walk produces is not unique #

One property worth recording, because it makes the walk’s output weaker than it might appear.

At each round the walk places everything it can, so its output is a sequence of rounds rather than a single ordering, and within a round the order is whatever the joint list happened to be. A different traversal would give a different sequence and the same verdict.

That is fine for the question being asked — does an order exist is not sensitive to which one — and it means the printed order is not a canonical object and should not be treated as one. The topology field is careful about this elsewhere: a canonical form is a thing two of which can be compared, and a construction order is not one, because two runs on the same machine can produce different sequences.

What is canonical is the verdict and the counts: complete or stuck, how many placed alone, how many dyads, and what the largest irreducible group is. Those do not depend on the traversal, and they are what this rung quotes.

Why the site solves it anyway #

If every joint can be placed in order, the Newton–Raphson this site positions everything with is doing work a compass and straightedge could do. It is worth saying why it is used regardless.

Construction order chooses a branch at every step and says nothing about which. Placing a joint from two circles means picking one of two intersections, and picking it by which one the construction meant is exactly the assumption that the branch rung shows is unwarranted. A constructed position is a position on the intended branch by definition, which makes it useless for asking whether a machine is on the intended branch.

A solve reports a residual. A construction reports nothing. The site’s invariant is that nothing is drawn that was not solved, and the reason is that a solve has a number attached saying how well it did.

So the construction is used once, to seed, and everything after it is a march in which the solver does real work from a nearby guess. The walk in this rung is a statement about the machine’s graph, not about how its positions are found.

What the braces do to the walk #

Bracing a compiled machine adds a bar and two rigid attachments per parallelogram, and it is worth asking whether the walk survives it.

It does, and easily. A brace’s two midpoints are rigid attachments to links already placed, so each is placed alone the moment its link is; the bar between them is then a redundant condition on two joints that already have positions. The walk never reaches it, because the walk stops caring about a joint once it is placed.

That is a small observation with a useful corollary: a redundant constraint is invisible to a construction order. The walk on a braced machine reports the same structure as on an unbraced one, because a construction order is about what determines what, and a constraint that determines nothing new determines nothing.

Which means the walk is exactly blind to the thing that makes the braced machine’s Grübler count come out at minus a hundred and fifty-three. Two instruments, two blindnesses: the count cannot see that its surplus equations are redundant, and the walk cannot see that they are there. Neither is wrong and neither is enough on its own.

One number the walk does not report #

A last limit, stated because the essay’s headline invites a stronger reading than the measurement supports.

Nothing larger than a dyad is a statement about the walk finishing with dyads as its largest step. It is not a proof that no larger Assur group exists in the machine, because the walk is greedy: it takes the first thing it can place, and a greedy walk that finishes has shown a decomposition exists rather than shown that no other decomposition is forced.

For this purpose the distinction does not bite. A machine that can be taken apart into dyads is a machine with no larger irreducible group — decomposability is a property of the chain, not of the order it was taken apart in — so the verdict stands. What would not stand is a claim in the other direction: a walk that stalled would not prove an irreducible group exists, only that this walk could not find a way past it.

The site has been caught by that asymmetry before, in the mobility instruments: a count that comes out negative is evidence of overconstraint and a count that comes out positive is evidence of nothing. Same shape, and the honest way to quote either is to say which direction it is evidence in.

What this adds to the topology field #

The field’s census counts chains and asks which are distinct, which are isomorphic, and which have rigid subchains. Its objects are small — ten links is a large census — and its interest is in structure.

This rung supplies a specimen from the far end of a scale the field has not had one from: a chain of four hundred links whose structure is completely trivial, produced not by search but by construction. It is a useful negative example. The census’s difficulty is not about size, and here is a mechanism sixty times larger than anything the census enumerates with less structure in it than the four-bar.

There is a second specimen hiding in it that the field might want. A compiled machine is a chain built from a vocabulary of four subgraphs — a two-bar dyad, a rhombus, a railed rhombus, a parallelogram — repeated hundreds of times and wired together in a tree. The field’s census enumerates chains without regard to how they were made, and a chain assembled from a small vocabulary is a different kind of object from one that happens to have the same graph: its automorphisms come from the repetition, and the orbits the census counts would be enormous.

Nothing here computes those, and it is recorded as a direction rather than a result. A census over chains built from a stated vocabulary is a smaller and more structured problem than the general one, and it is the shape of problem a construction like this one poses.

It also supplies a check the field could use elsewhere. The walk is fifteen lines and it decides, for any mechanism given as joints and constraints, whether a construction order exists. The field’s existing test for the same thing is Assur decomposition, which is a different algorithm answering the same question — two routes, and the site has only ever had one.