The chain has no lengths

A census is a list of graphs and a graph has no shape. That has been stated in every rung of this field and it is worth ending on, because it is the boundary of everything the field claims and because the figures make it easy to forget.

The test that separates them #

There is a clean one and it applies to every quantity in the field.

Multiply every link in a mechanism by a different scale factor. Not the whole mechanism by one factor — each link by its own. The result is a different mechanism entirely: different coupler curve, different transmission angle, different workspace, possibly a rocker where there was a crank.

Every number this field computes is unchanged, because there was nothing to multiply. The census counts do not move. The assortment does not move. How many mechanisms a chain gives does not move. Whether it comes apart into pairs does not move. The characteristic polynomial does not move.

That test is the field’s whole boundary, and it is the same shape as the boundary the holding field drew against force: every number there survives with every force unknown, and every number here survives with every length unknown.

What the graph does decide #

Worth listing, because it is more than a reader would guess.

How many degrees of freedom the count claims, which is two integers. Whether the graph is a mechanism at all, which is the subset scan. How many mechanisms it gives, which is an orbit count. How many genuinely different ways there are to drive it. Whether it can be positioned two links at a time, and therefore whether a closed form exists. How many assemblies a dyadic decomposition has, which is two to the power of the dyads. How many independent loops there are, and the smallest number of link lengths any one of them can involve.

That is a substantial list and several of the items are things a designer would otherwise assume are dimensional. In particular, the number of assembly branches is topological — a Watt six-bar has four assemblies at a given input angle whatever its lengths are, provided it assembles at all.

What it does not #

A longer list, and every item on it is something a census will be asked for by somebody who has just seen one.

Whether the driven link turns fully. Grashof’s condition compares four numbers and a chain has none. The chart at the top of this essay puts ten of the sixteen chains at a full revolution and six between 134° and 244°, and every one of those bars would move if the placement did.

What the coupler traces. A coupler curve is a sextic whose coefficients are the link lengths; the graph decides only that there is one.

Whether it assembles at all. Four lengths can fail to close a loop, and then there is no mechanism to speak of. The census counts chains that can be given dimensions, not ones that have been.

Where the singularities are. Toggle positions and dead centres are configurations, so they need both a set of lengths and a place in the motion.

How accurate it is. Which link lengths matter most is a sensitivity, computed from a Jacobian at a configuration, and what a stack-up does with it needs numbers this field does not have.

And whether it is any good. Every criterion an engineer would use — smooth motion, adequate transmission angle, a workspace covering what is needed, a mechanical advantage that does not collapse — is dimensional, and a census ranks nothing.

Where the site’s other fields sit relative to this line #

It is worth placing the whole collection on the boundary, because almost every field is on the other side of it and a reader arriving here will want to know what they have been reading.

The linkages, curves, curvature, synthesis and algebra fields are entirely dimensional. Every quantity in them — a coupler curve, an inflection circle, a Burmester point, a root count — is computed from lengths, and moving a length moves it. The census has no opinion about any of them.

The constraint and spatial fields are the closest to this one, because a mobility count is a count and a Jacobian rank is nearly one. That closeness is exactly what makes them the fields this one corrects: they compute the same quantity by two routes and neither route can see a rigid subchain.

The networks field is closer still, and interestingly so. Its assemblies are parameterised by a size — a scissor chain of $n$ units, a sheet of $n \times n$ panels — so its quantities are functions of a count rather than of a length, and several of them are genuinely topological. What separates it is that its counts are about repetition and this field’s are about arrangement: a Miura sheet of any size is one pattern repeated, and a ten-link chain is thirteen pins placed one way out of 1,878.

And the holding field is the other one whose numbers survive a change of something — every one of them survives with every force unknown. The two fields draw their boundaries against different things and by the same test.

The layout is arbitrary and reproducible #

A last point about the figures, since the field’s honesty depends on it.

A graph has no geometry, so where the discs go is a choice, and this field makes that choice once, mechanically, from the canonical form. Links start on a circle in canonical order, relax under a fixed schedule, and the result is rotated and reflected by a fixed rule. There is no seed, no hand-placement and no per-chain adjustment.

That has two consequences and both matter here. The same chain always draws the same picture, whatever labelling it arrived in — checked by relabelling forty times at random. And the picture means nothing: the distance between two discs is an artefact of a relaxation, not a length, and no figure in this field measures anything off one.

The mechanism drawings inherit the same arbitrariness one step further along. Their pin positions come from the graph layout, so the lengths that result are a consequence of a spring simulation and carry no design intent whatever. That is why the arc chart is the way it is, and why it is presented as a warning rather than as a result.

How the figures keep it honest #

This creates a genuine problem for a field whose figures have to show mechanisms sometimes, and the way it is handled is worth stating because the alternative is quietly misleading.

A chain is drawn as a graph — discs and lines, no geometry — wherever the argument is about the chain. That is most of the field.

Where a mechanism is drawn, the pins are placed first and the links are read off them. A pin is a point of the plane and both its links pass through it, so the assembly is exact by construction: the closure residual is nought rather than small. The lengths that result are whatever the graph layout produced, which is to say arbitrary.

And every caption says which. A figure showing a driven mechanism says that the lengths are a representative, that the arc belongs to the placement, and that the census above it does not change. That is repetitive, and it is repetitive on purpose: a reader who has looked at twenty pictures of mechanisms is entitled to think the field is about mechanisms.

The one place they meet #

There is a single quantity in this field computed from a solve rather than from a graph, and it is here to mark the boundary rather than to cross it.

The arc chart at the top is that quantity. Each of the sixteen chains is given the placement its own layout produces, driven from its first available frame-and-input pair, and swept by continuation until a frame stops closing. The numbers that come out — 134°, 179°, 244°, a full turn — are properties of sixteen arbitrary mechanisms.

It is in the field because the temptation to read a topology census as a catalogue of machines is strong, and the sharpest correction is a chart of numbers that look like they belong to the census and do not. Ten of sixteen reaching a full turn is not a fact about eight-link chains; it is a fact about one layout algorithm.

An objection worth taking seriously #

The honest objection to a whole field of this is that a designer never chooses a topology from a list, and that the census is therefore an answer to a question nobody asks.

It has force. The two-stage division — type first, then dimensions — is textbook and is not how a working engineer proceeds. What actually happens is that a mechanism is adapted from one that worked, or grown by adding a dyad to something that nearly worked, and the topology is a by-product of that process rather than a decision in it.

Two answers, and only the second is strong.

The weak one is that the process could be improved, which is easy to say and hard to demonstrate. A designer’s repertoire is drawn from things that have worked, and things that have worked are a better prior than a uniform distribution over 230 graphs.

The strong one is that the census makes negative statements available and nothing else does. There is no third six-link chain. No ten-link mechanism has a link with six pins. These four chains have no closed-form driving choice from any of their eighty driving pairs. Each of those is a claim about the whole of a space, each is the kind of claim a design review needs and cannot otherwise get, and none of them could be reached by adapting a mechanism that worked.

That is the argument for the field, and it is deliberately narrow. It does not claim that enumeration designs anything better; it claims that some questions have exhaustive answers and that having them is worth half a second.

Generic counts belong to the graph and real ones do not #

Two items on the list of things a graph decides sit oddly there, and they are the two most worth understanding, because they are the ones a designer would swear were dimensional.

How many assemblies a mechanism has feels dimensional. Two circles meet in two points, or in one, or in none, and which of those happens plainly depends on their radii — so how can a count of assemblies belong to a graph? And whether the position problem has a closed form feels dimensional for the same reason: surely a special arrangement of lengths could make a hard system factor.

The resolution is one distinction and it runs through the whole of the algebra field. The graph decides the count generically — over all dimensions at once, with the degenerate cases set aside — and the dimensions decide which of the generic solutions are real and assemblable at a particular set of lengths. Two circles always meet in two points; whether those points have real coordinates is what the radii decide. A four-link Assur group’s eliminant always has its degree; whether the roots are real is dimensional.

So the two lists are not divided by which quantities the lengths can touch. They are divided by whether the quantity is a count over the whole family or a fact about one member. A census can say that a chain solved as two dyads has four assemblies, and it cannot say that this four-bar with these four lengths can be put together in four ways — because at some lengths two of the four are complex and the mechanism has two, and at others it does not close at all.

That is a sharper statement of the boundary than the graph knows no lengths, and it explains why the boundary falls where it does rather than merely reporting that it falls there. Everything on the “decides” list is a count or a structure that survives being averaged over the whole space of dimensions. Everything on the “does not decide” list is a property of a point in that space: whether this crank turns, what this coupler traces, whether these four lengths close, where this mechanism’s dead centres are.

It also says exactly how a census result should be quoted, which matters because the sentence is easy to write carelessly. A Watt six-bar has four assemblies is a claim about the chain and is true generically. This Watt six-bar has four assemblies is a claim about a mechanism and needs the lengths. The census supports the first and is silent about the second, and the difference between them is a single demonstrative pronoun — which is a poor place for the whole boundary of a field to live, and is why the field says it out loud.

Why the boundary is worth defending #

Two reasons, and the second is the one that matters.

The first is that a census over-claims easily. There are 230 ten-link mechanisms is a sentence somebody will write, and it is wrong twice over — there are 1,834, if a mechanism is a chain with a frame, and there are infinitely many if a mechanism has dimensions. Being precise about which object is being counted is most of what makes the counts useful.

The second is that the separation is what makes the field’s results transferable. A result about a graph applies to every mechanism with that graph, at every set of dimensions, for ever. This chain has no closed-form driving choice is not a fact about one linkage somebody built; it is a fact about all of them, including the ones nobody has built yet.

That is the trade the field makes, and it is a good one. It says less about any particular mechanism than any other field on this site, and what it does say is true of every mechanism at once.

What a reader should take from the field #

Five things, and none of them is a number.

A count of anything is a filter, not an answer. Grübler’s rule admits eight graphs for every mechanism at ten links, and it admits them at generic dimensions with no special geometry to blame. Every arithmetic condition in the subject has this character: necessary, cheap, and much weaker than it looks.

Two routes agreeing is evidence about the quantity they compute. The site’s standing habit caught nothing here, because both of its routes compute the mobility of the whole assembly and the question was about a subset. The useful discipline is to ask what quantity a pair of routes agrees about and whether it is the one in the claim.

A test that is exact on every case anybody checked is a test checked up to a size. The spectral fingerprint separates every chain at six and eight links and fails at ten, which is one row past where hand-verification stops. That coincidence is not a coincidence.

A solver is sometimes the only route. Four of the sixteen eight-link chains and ninety of the 230 ten-link ones have no closed-form driving choice at all, so the Newton solve this site has run on everything is not a convenience there. It is the only thing that produces a position.

And the discrete decisions come first and are never revisited. Link count, chain, frame, input: four choices, all finite, all made before the optimiser starts, and all of them invisible to it afterwards.

There is a mirror of the same distinction on the other side of the boundary, and naming it keeps the trade honest. Just as the graph’s counts are generic and say nothing about a particular mechanism, a dimensional result is about a particular mechanism and says nothing about the family. A measurement that this four-bar’s transmission angle never falls below forty degrees is exact, checkable, and carries no information whatever about the next four-bar. Neither side of the line is the more useful one; they answer questions of different shapes, and the reason the two-stage workflow exists is that a designer needs both answers and cannot get either from the other.