Four that a compass cannot reach

A mechanism whose Assur groups are all dyads can be positioned with a compass: two circles, two branches, in a fixed order, exactly. Everything the classical repertoire contains is of that kind — every four-bar, every slider-crank, every Watt six-bar, and eight of Stephenson’s fourteen driving choices.

The natural inference is that a mechanism which resists is being driven from the wrong link, and that some other choice of frame and input would let it come apart. On most chains that inference is right.

On four of the sixteen eight-link chains it is wrong, and it is wrong exhaustively.

The count #

Every chain has $2j$ frame-and-input pairs — twenty at eight links, since $j = 10$ — and rather fewer distinct ones. Running the decomposition on all twenty of all sixteen chains gives 320 answers, of which 150 are all dyads.

Those 150 are not spread evenly. Twelve chains have at least one all-dyadic choice and four have none.

At six links the answer is nought: both chains have dyadic choices, Watt’s from all fourteen and Stephenson’s from eight. At ten links it is ninety of the two hundred and thirty — nearly two in five — so the exceptional case stops being exceptional exactly one census row after it first appears.

What the four have in common #

Not much, which is the honest answer and the interesting one.

Three of them are in the assortment with four ternary links and one has a quaternary, so it is not the assortment. Their automorphism groups have 16, 8, 8 and 2 members — three of them are among the most symmetric chains in the census, against a median of three across the sixteen, and the fourth is exactly median.

The tendency has an explanation. A chain with many automorphisms has few genuinely different places to attach a driven link — the most symmetric of the four has twenty listed pairs and very few distinct ones — so it gets fewer independent attempts at a good decomposition. That is a statement about how many chances a chain gets, not about whether any of them could have worked.

The real obstruction is about loops sharing links rather than pins, which is the same fact that separated Watt’s chain from Stephenson’s. In these four the loops overlap heavily enough that no pair of links ever has both of its outer pins on placed parts, from any starting point.

What “no closed form” means and does not mean #

Three clarifications, because the phrase invites over-reading in both directions.

It does not mean the mechanism cannot be positioned. These four chains have solvable position problems, in the sense the constraint field’s opening rung means it. The solve converges, the assembly closes to a part in $10^{13}$, and the mechanism can be swept and drawn like any other. What is absent is a construction: a finite sequence of circle intersections that produces the answer without iterating.

It does not mean the position problem has no algebraic solution. Eliminating variables from the loop-closure equations of a four-link group gives a polynomial in one unknown, and that polynomial has roots. What it does not have is a factorisation into quadratics, so the roots are not obtained by intersecting circles and there are more than two of them. That is the algebra field’s ground and its instruments are total degree, homotopy continuation and root counting rather than a compass.

And it does not mean the mechanism is worse. Nothing about a chain’s solvability class says anything about its motion, its transmission angle, its workspace or its accuracy. It is a statement about the cost of computing where it goes, which in 1890 was the binding constraint and today is not.

Changing the input moves the problem and does not remove it #

The clearest way to see that the obstruction is real is to watch what happens when the driving choice is changed on one of the four.

Take the least symmetric of them, which has six distinct mechanisms and therefore several genuinely different places to attach a crank. Ground a ternary link and drive one of its neighbours: the decomposition returns a group of four and then a group of two. Ground a binary link instead: a group of six, all at once. Ground the other ternary link: four then two again, but a different four.

Across all twenty choices the group sizes move around and the largest is never two. That is what “exhaustively” means here: not that the natural choice failed, but that the search over choices was run and returned nothing.

It is worth contrasting with the twelve that do work, because several of them are only just on the right side of the line. Two of the sixteen chains are dyadic from all twenty of their choices, four more from fourteen of twenty, and several from eight of twenty — so a designer picking a driving link by convenience on one of those has a real chance of landing on a non-dyadic choice and concluding, wrongly, that the chain is one of the hard ones.

What replaces the construction #

Since these mechanisms have to be solved rather than constructed, it is worth being explicit about what the solve actually is on this site, because it is not the loop-closure formulation the rest of the collection uses.

The unknowns here are pin coordinates, two per joint. The equations are: for every link, the distance between each pair of its own pins, held at whatever the placement made it; the frame’s pins, held where they are; and the driven link’s angle. A link with four pins contributes six distance equations of which only five are independent, so the system is deliberately over-determined and solved by damped least squares.

Writing the independent subset instead would mean choosing one, and choosing one is a place for a bug to live — which is the same decision the networks field made for the same reason.

There is one trap in that formulation and it cost this field a wrong answer before it was caught. Three distances fix a triangle only up to reflection, so a link carrying three or more pins has a mirror image satisfying every equation exactly, and a solve from a cold seed lands on it about half the time. A reflected link is not a pose of the mechanism — it is a different part — and drawing one would be a figure showing a configuration the mechanism cannot reach.

A four-bar never shows it, because every link is binary and two points have no handedness. Every mechanism in this field with a ternary link would have. The reference placement fixes the sign, continuation preserves it, and a frame that has flipped is now reported as unconverged rather than drawn.

Why it mattered, and why it stopped mattering #

The historical weight of this distinction is easy to miss from a position where a solver is free.

Before numerical computation, a mechanism that could not be constructed could not be drawn — not in the sense of drawn badly, but in the sense that there was no procedure to find where its links went. Graphical kinematics was the whole discipline: instant centres, velocity polygons, Burmester’s constructions, and every one of them a ruler-and-compass method that assumed the mechanism came apart into pieces the compass could handle.

So a designer’s repertoire was shaped by what could be constructed, and the four chains above — and the ninety at ten links — were effectively invisible. Not rejected: unavailable. A search that cannot evaluate a candidate does not report it as bad.

That constraint is gone, and this rung’s real content is what replaced it. The census is now searchable in full, and whether a candidate is dyadic is a filter a designer might apply for other reasons — a closed form is faster, has enumerable branches, and is differentiable in the link lengths without an implicit-function argument — rather than a boundary on what can be considered at all.

What a solver has to do differently #

There is one practical consequence and it is the reason this essay sits next to the sweeps in this field.

On a dyadic mechanism the branches are enumerable. Each dyad has two solutions, the choices are independent, and a solver can be told which branch to take by naming a sign at each step. Following a mechanism through a motion means keeping the signs fixed and watching for the configurations where two circles stop meeting.

On a four-link group there is no such handle. The branches are roots of a polynomial nobody has factored, the solve arrives at whichever one its seed was nearest, and there is no sign to fix. The only reliable way to stay on one branch through a sweep is continuation: solve at a nearby parameter value, use that answer as the seed for the next, and never solve cold.

That is exactly what every drag figure in this field does, and it is why the frames of a slider here are sampled from a walk rather than solved independently. A cold solve at the far end of Stephenson’s arc fails; the continuation reaches it comfortably.

What is lost when the branches stop being countable #

The cost of a four-link group is usually described as speed, and speed is the least of it. What actually goes is the ability to say how many answers there are.

On a dyadic chain the assembly count is settled before any solving happens. Each dyad contributes two circle intersections, the choices are independent, and the number of assemblies is two to the power of the dyads — a count read off the decomposition rather than found by looking. A solve that returns fewer has missed some; one that returns more has found something that is not an assembly, which is exactly how the mirrored ternary link above was caught. The count works as an instrument because it comes from the structure and not from the search.

On these four chains there is no such number. The four-link group’s eliminant is a polynomial whose degree is not established here and whose roots are found by seeding a solve and seeing where it lands. Run it from a thousand random seeds and some set of distinct converged configurations comes back; run it from a thousand more and the set may grow. Nothing in the procedure says when to stop, because nothing in it knows what it is looking for.

Three things follow, and each changes what can honestly be claimed.

A completeness claim becomes a sampling claim. This chain has eight assemblies is a theorem on a dyadic mechanism and a report on one of these. The honest form of the second is eight were found, with the seeding scheme attached — and the difference is not pedantry, because a branch that is hard to reach from random seeds is precisely the branch nobody has considered.

Branch tracking loses its guard rail. A dyadic sweep carries a sign at each dyad, so a frame that has jumped is detectable: the sign flipped, and the frame is rejected rather than drawn. A continuation through a four-link group carries nothing comparable. If two roots approach and exchange, the solve follows whichever is nearer and reports the residual it reports everywhere else. The sweep looks clean and the mechanism has changed assembly halfway along it — the one failure this field’s continuation habit is designed to prevent, with no second line of defence behind it.

And a search over branches loses its denominator. Choosing a mechanism often means choosing an assembly: the branch on which the transmission angle stays open, or the one that reaches the far pose. On a dyadic chain that is a finite enumeration and the best of them can be named as the best. Here it is an optimisation over a set of unknown size, and every answer carries the qualification the assembly count carries.

The severity is worth keeping in proportion. None of this makes the four chains unusable, and the ninety at ten links are not a class to be avoided — a mechanism does not move worse because its position problem is harder to certify. What is lost is a particular kind of confidence, and it is lost at the size where it would be most useful: the census gets large exactly when the arithmetic that made it checkable stops applying.

There is a mild irony in how the four were found. The decomposition that fails on them is the same one that would have counted their assemblies had it succeeded, so the procedure reports its own inapplicability: it returns a group of four, and a group of four is simultaneously the reason there is no construction and the reason there is no count. A census of solvability and a census of countable assemblies are the same census, taken once. That is why the four appear in this field rather than in the algebra field, where the polynomial and its roots properly live: what is being reported is not a hard equation but the absence of a structural shortcut, and absence is a property of the chain rather than of its dimensions.

That is also why the assembly count is worth an essay of its own rather than a paragraph. It is not a formula for a quantity that always exists; it is the one case in which the quantity is knowable at all, and knowing where it stops is most of what it is for.

The honest limits of this count #

Two, and both are about what the decomposition assumes.

The input is a crank. Every count here assumes the driven link is pinned to the frame and turned about that pin. Drive a mechanism with a linear actuator between two arbitrary links and the known set at the start is different, the arithmetic is different, and a chain with no dyadic crank choice may well have a dyadic actuator choice. Nothing in this essay rules that out and nothing in it has been computed.

And “all dyads” is a sufficient condition for a construction, not a necessary one, in the way a bound is not a count. There may be constructions for some four-link groups that this arithmetic does not see — a special arrangement of pins, a case that factors for a structural reason. The claim made here is precise and narrower than the headline: no order exists in which these chains come apart two links at a time. That is a statement about the decomposition, and it is what the figures measure.

What it says about the site’s own habit #

There is a closing observation about method rather than about mechanisms.

This site has run a Newton solve on every mechanism it has ever drawn, and the justification has always been generality: nothing is drawn that was not solved, so a figure cannot show a configuration the mechanism cannot reach. That is a statement about trust, and it has always been possible to read the solver as an expensive way of getting answers a construction could have given.

This rung says the reading is wrong on four of the sixteen eight-link chains and on ninety of the two hundred and thirty ten-link ones. On those, there is no construction to have used. The solver is not an implementation choice; it is the only thing that can produce a position at all, and a site that had chosen constructions would have had nothing to draw.

That is worth having as a defence of a method that usually looks like belt and braces. A general instrument is justified by the cases it reaches and not by the cases where it agrees with something cheaper — and the cases it reaches are not visible until somebody enumerates them. Before this field the site could say that its solver was more general than a construction. It can now say how much more general, at each size, as a fraction.