The mechanism is the graph

Every field on this site so far starts with a mechanism. The four-bar arrives with four links and four pins already decided; a Gough platform arrives with its six legs; a Miura sheet arrives with its creases. The question is always geometric — where does it go, how fast, how many assemblies, how far out of true — and the answer is always a solve.

Take the geometry away. Keep only which link is pinned to which, and throw out every length, every angle and every position. What is left is a graph: a dot for each link, a line for each pin. It has no shape, it cannot be drawn in a position because it has no positions, and nothing about it can be measured with a ruler.

It is still the object almost every claim in the subject is about. Grübler’s rule reads two numbers off this picture. Whether a mechanism is a four-bar or a six-bar is a statement about it. Which links can be grounded, whether the position problem has a closed form, how many assembly configurations there are, how many independent loop equations a tolerance analysis must satisfy — every one of those is decided here, before a single dimension is chosen.

So the question this field asks is the one the rest of the site takes as given. Which graphs are mechanisms, and how many are there?

What counts as a chain #

Four conditions, and each of them is a modelling decision rather than a convention.

Simple. Two links joined by two separate pins cannot move relative to one another at all. If a graph has a repeated edge it is describing one rigid link drawn as two, so repeated edges are out.

Every link carries at least two pins. A link with one pin is a flag: it swings, and nothing beyond it does. Excluding it is what makes the count finite in a useful way — otherwise any chain plus a dangling stub is a new chain.

Connected. Two components are two mechanisms and the question is about one.

No subchain is already a structure, which is the condition that does the work and gets an essay of its own. Take any subset of the links with the pins that run between them: it is a small chain in its own right, and it has its own count. If that count is nought, the subset never moves internally, and the whole thing is not a mechanism with this many links.

Mobility is planar throughout: a link in the plane has three freedoms, a pin takes two away, one link is held still, so $M = 3(n-1) - 2j$. Setting that to one forces $j = (3n-4)/2$, which is an integer only for even $n$. That is why the census has rows at four, six, eight and ten links and no rows between them — the arithmetic itself refuses odd link counts before any graph is looked at.

One, two, sixteen, two hundred and thirty #

With the four conditions stated, the census is a finite search and the answers are integers.

Four links, one chain. There is exactly one graph on four vertices with four edges and every vertex of degree two, and it is the four-bar. Everything the linkages field has ever drawn — crank-rocker, double-crank, double-rocker, the slider-crank as a limit — is that one graph, with different lengths and different links held still.

Six links, two chains. Both have four binary links and two ternary ones; they differ in whether the two ternary links share a pin. Watt’s does and Stephenson’s does not, and that single fact is the whole difference between them.

Eight links, sixteen chains. This is where the subject stops being enumerable by hand and starts being enumerable by machine. Nine of the sixteen have four ternary links, five have two ternaries and a quaternary, and two have a pair of quaternaries.

Ten links, two hundred and thirty. And that number is the one worth pausing on, because it is a number the literature has carried since the mid-twentieth century and it comes out here from a search written for this site, in about half a second, with no table consulted.

The count admits eight graphs for every one that deserves it #

The last column of the census table and the one before it are not the same question, and the gap between them is this field’s first result.

At ten links, 1,878 graphs satisfy every arithmetic test anybody would apply: connected, simple, ten links, thirteen pins, no link with fewer than two pins, mobility exactly one by Grübler’s rule. 230 of them are mechanisms with ten links. The other 1,648 carry a subchain that is already a structure.

The smallest case is small enough to check by eye. At six links, five graphs pass the count and two are chains; the other three each contain a triangle, and a triangle of three binary links is the smallest structure there is — $3 \times 2 - 2 \times 3 = 0$.

This is not a technicality about how to name things. A designer who searched a list of 1,878 topologies would find most of them describing mechanisms they had already considered, in a disguise that a count cannot lift. And it is not a small correction that matters at four links and washes out at ten. The ratio goes the other way: 1.00 at four links, 2.50 at six, 4.44 at eight and 8.17 at ten, roughly doubling every step. The fourth condition does more of the work the larger the census gets.

The coarsest thing that can tell two chains apart #

Before any search over relabellings, there is one number per link that costs nothing: how many pins it carries. Two, three, four. The multiset of those — the link assortment — is what every published census table is organised by, and it is the first test anybody applies, because two chains with different assortments are certainly different chains.

The assortments themselves come out of two lines of arithmetic: the degrees must sum to twice the pin count, and none may be below two. At ten links that admits eleven. Four of the eleven contain no mechanism whatever — every graph with those degrees, and there are seventy-eight of them, has a structure inside it. That is a fact the arithmetic cannot reach, because the arithmetic never looks at where a pin goes; it takes an enumeration to find, and it is the first hint that this field’s questions do not answer to counting.

And the assortment is emphatically not enough. Watt’s chain and Stephenson’s have the identical assortment — four binary links and two ternary — so the cheapest test passes them as equal, and they are the two different six-bars every textbook names separately. The next rung is about exactly that gap.

A chain is not a mechanism until a link is held still #

The rightmost column of the census is larger than the one beside it, and the reason is that a chain is not yet a machine.

A mechanism is a chain plus a decision about which link is bolted to the floor. Grounding different links of the same chain gives different machines: the same graph, the same pins, and a completely different relationship between what turns and what moves. The linkages field made this argument for the four-bar and the serial field made it for a robot’s wrist; here it becomes a count.

Two links give the same mechanism when some relabelling of the whole chain carries one to the other and leaves every pin where it was. So the number of genuinely different mechanisms a chain gives is the number of orbits of its links under its own symmetries.

Watt’s chain has two orbits, Stephenson’s has three, so there are five six-bar mechanisms. The sixteen eight-link chains have seventy-one orbits between them, so there are seventy-one eight-link mechanisms. Both numbers are classical, and both are here a count of orbits rather than a list somebody compiled.

Nothing here has a length, and that is the point #

It is worth being blunt about what this field cannot do, because a census of topologies looks like a catalogue of machines and is not one.

Give the graph some lengths and it becomes a mechanism that can be driven, and everything the rest of this site measures becomes available: whether the crank turns fully, how good the transmission angle is, what curve the coupler draws. None of it is decided here.

The separating test is a clean one and it is worth stating once for the whole field. Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. A census count does not change. A chain’s assortment does not change. How many mechanisms it gives does not change. Whether it can be positioned in closed form does not change. What does change is everything the previous twenty-one fields measure.

Why a picture of a graph is a problem #

There is one difficulty peculiar to drawing this field, and it had to be solved before any figure above could be trusted.

A graph has no geometry, so where the discs go on the page is a choice. Two drawings of the same chain that looked different would be the worst thing this family of figures could do, because are these two the same chain is the central question of the field and the reader’s first instrument is their eye.

The rule the figures are built on is therefore mechanical: the layout is a function of the chain’s canonical form and of nothing else. Every chain relaxes from a circle whose order comes from its canonical labelling, by a fixed number of steps with a fixed schedule, and the result is rotated so its longest axis is horizontal. A chain arriving under any labelling whatever draws the same picture, which is checked by relabelling one forty times at random and requiring the drawing to be unchanged.

That is the same discipline the rest of the site applies to positions — nothing is drawn that was not solved — applied to the one thing here that has no solve behind it.

What the census is for #

A census is not a curiosity, and the argument for having one is not that the numbers are pretty.

It bounds a search. Dimensional synthesis — given the motion, find the lengths — is a search over shapes within a topology, and it is started by choosing a topology, usually from memory and usually from a list of five. With the census the list is 230 at ten links, and a requirement stated about the graph alone cuts it before any dimension is chosen.

It tells a solver what it is up against. Whether a mechanism can be positioned two links at a time with a compass, or has to be solved as one system, is decided by the graph — and four of the sixteen eight-link chains cannot be positioned by any construction from any choice of frame and input.

And it says what the counting rule is worth. Grübler’s rule has been on this site since its first field, and the known failure has always been about geometry: a formula that says a mechanism cannot move while it demonstrably does, because the link lengths are special. This field finds a second and larger failure that has nothing to do with geometry at all. At ten links the rule admits eight graphs for every mechanism, at generic dimensions, with no special lengths anywhere — and the instrument that catches it is neither the count nor the rank the site has been checking the count against.

That is the next rung, and it is where this field earns its place.

What each condition is protecting against #

The four conditions are called modelling decisions above, and it is worth taking them one at a time and asking what the census would be counting if each were dropped. Each answer is a different object with a different size, which is the clearest way to see that the four are doing four distinct jobs rather than tidying one.

Drop simplicity and two links may be joined by two pins. That arrangement cannot move — two pins fix the relative position of two bodies completely — so a multigraph of this kind describes a mechanism with one fewer link than it claims, its two doubly-pinned links being one rigid body. The census would count welded assemblies alongside mechanisms, and it would count each one many times over, since any pair of links can be welded in this way.

Drop the minimum degree and links may carry one pin. Such a link swings freely and transmits nothing; it is a flag on a mechanism rather than part of it. The census would then be infinite in spirit, since a flag can be hung on any link of any chain and the count would grow without any new mechanism appearing.

Drop connectivity and the census counts sets of mechanisms. Two four-bars side by side would be a graph of eight links satisfying the count, and it would sit in the eight-link row beside the genuine eight-link chains. The number would be right about something and about nothing anybody asked.

Drop the structure condition and the census counts 1,878 at ten links instead of 230, which is the gap this field’s second rung is about. What comes in is not rubbish but duplication: graphs describing smaller mechanisms with rigid subchains written out as several links, so the four-bar appears repeatedly wearing extra triangles.

Read together, the four failures are of three kinds. Two of them — simplicity and the structure condition — admit graphs that describe smaller mechanisms, so the census inflates by duplication. One of them, the degree condition, admits graphs that describe the same mechanism with decoration attached. And one, connectivity, admits graphs that are not mechanisms at all.

That is why the four are stated as conditions on the object rather than as filters on a search. Each one is the statement that a particular way of writing down a mechanism with more links than it has, or with parts that do nothing, is not a different mechanism. Every census in this field is a count of mechanisms up to those four identifications, and the numbers mean nothing without them.

How the rest of the field goes #

The ladder from here has a shape worth stating, so a reader can see where each rung is going.

The next two rungs are the object itself: the two six-link chains that no count can separate, and the gap between the graphs a count admits and the chains that deserve it. Then two on how the question is actually answered — the spectral test that is exact on every census small enough to check by hand and wrong at ten links, and the canonical form that replaces it.

Then two rungs on mechanisms rather than chains — which links may be held still, and what symmetry costs a chain in machines. Then two on the position problem: which sets of links have to be solved together, and the four eight-link chains for which no compass construction exists from any driving choice. Then two on the search itself — what a census costs and where it stops — and the assortment table above read properly. And last, two rungs on what the census is for and what it cannot say: a catalogue as a search space, and the chain that has no lengths.

Four essays outside the field draw on it. The constraint field gets a second way for its count to be wrong. The synthesis field gets the stage before dimensional synthesis. The algebra field gets a prediction of how many assemblies a chain has, made from the graph. And the spatial field gets the reason every spatial mechanism on this site is a single loop, which turns out to be a census of size one.