In space there is one chain

The spatial field opens by changing two numbers. A body in space has six freedoms rather than three, and a revolute joint takes five rather than two, so Kutzbach’s count replaces Grübler’s:

$$M = 6(n-1) - 5j.$$

Every consequence the field draws from that is about particular mechanisms — Sarrus’s linkage counts as immobile and gives exact straight-line motion, Bennett’s four-bar moves only for one combination of lengths and twists, a planar four-bar is over-constrained too when the spatial count is applied to it.

Ask the same question the topology field asks of the plane — which graphs are mechanisms, and how many are there — and the answer is startlingly small.

Only three link counts are possible #

Setting $M = 1$ gives $j = (6n-7)/5$, and a joint count has to be an integer. That happens only when $6n - 7$ is a multiple of five, which is to say when $n$ leaves a remainder of two on division by five.

So the admissible sizes are 7, 12, 17, and then every fifth number after. There is no spatial all-revolute mechanism of one degree of freedom with eight links, or nine, or ten, or eleven — not a rare one, none, and the arithmetic says so before any graph is considered.

That is a much harsher filter than the planar one, which admits every even link count. Six freedoms per body and five per joint are nearly equal, so each additional joint buys back almost all of what an additional link costs — and the arithmetic closes only on a sparse set.

Seven links, and every one of them binary #

At seven links the joint count is seven, so the degrees sum to fourteen across seven links with none below two.

There is exactly one way to do that: every link carries exactly two joints. A graph in which every vertex has degree two is a disjoint union of cycles, and a connected one is a single cycle.

So there is exactly one spatial chain of seven links, and it is a seven-bar closed loop. Its automorphism group has fourteen members — the cycle’s rotations and reflections — and its links are all in one orbit, so it gives exactly one mechanism.

That is a complete answer to a question the spatial field never asked. Every spatial mechanism on this site is a single closed loop: the universal joint, Bennett’s linkage, Sarrus’s, the 7R loop the field’s counting essay uses as its one honest case. It has always read as a modelling convenience — single loops are what the field’s screw machinery handles — and it is not. At the smallest admissible size there is nothing else to build.

Where the over-constrained mechanisms went #

That last figure is the objection, and it deserves a straight answer.

Sarrus’s linkage has six links and six revolute joints. Kutzbach gives $6 \times 5 - 5 \times 6 = 0$, and it moves. Bennett’s has four links and four joints, giving $-2$, and it moves. Neither is in this census, and neither could be: the census selects on $M = 1$, and both of them count differently.

That is not a gap in the census; it is the census’s boundary, and it is the same boundary the whole topology field draws. A mechanism whose mobility comes from special dimensions is invisible to a count, and this census is built entirely on counts. The 7R loop is in it because its mobility is generic — it moves for almost any lengths and twists — and Bennett’s is not because its does not.

So the spatial census counts the mechanisms whose motion is a consequence of the connections. It is a small set, and the field’s most interesting objects are all outside it, which is worth stating plainly rather than glossing.

Twelve links, and five chains #

The next admissible size is twelve links and thirteen joints, and the arithmetic there admits two assortments: ten binary links with two ternary, or eleven binary with one quaternary.

The search reaches 157 candidates, finds 33 distinct graphs satisfying the count, and five of them are chains. Every one of the five has the ten-binary-two-ternary assortment; the assortment with a quaternary link contains no mechanism at all, exactly as four of the planar assortments do.

Each of the five carries two independent loops, which is $13 - 12 + 1$ and is forced. Their automorphism groups have 4, 2, 2, 4 and 4 members, giving 4, 7, 7, 4 and 5 inversions — twenty-seven spatial twelve-link mechanisms in total.

Five against 230 is the whole difference between the two subjects, stated as a ratio. The spatial arithmetic is so tight that almost nothing survives it, which is why the spatial field is about individual named linkages and the planar fields are about families.

Reading the seven-cycle properly #

The one spatial chain deserves a closer look, because a seven-bar loop sounds like an odd object and it is the most-studied linkage in spatial kinematics.

Seven links, seven revolute joints, in a closed ring. Kutzbach gives $6 \times 6 - 5 \times 7 = 1$, and unlike Bennett’s linkage or Sarrus’s the count is right: a 7R loop moves for almost any choice of link lengths, offsets and twist angles. Its mobility is a consequence of the connections rather than of a condition on the dimensions.

That genericity is exactly what makes it the field’s honest case and the field’s hard case at once. Its position problem — given one joint angle, find the other six — has no closed form of the dyad kind, and the classical answer is a polynomial of degree sixteen in a half-angle tangent. Sixteen assemblies, none of them constructible.

So the spatial census’s single answer is also the spatial field’s hardest position problem, and there is nothing else at that size to be easier. That is a sharper statement than spatial mechanisms are difficult: at the smallest size where a generic spatial mechanism exists at all, the only one there is is the difficult one.

What the ratio says about the two fields #

It is worth drawing the contrast out, because it explains a difference in style between two fields of this site that a reader might otherwise read as an accident.

The planar fields are about families. A four-bar is one of a class; a six-bar is one of five; an eight-link mechanism is one of seventy-one. Arguments are made about what happens across a family, and a census is the natural instrument.

The spatial field is about individuals. Bennett’s linkage, Sarrus’s linkage, the universal joint, the 7R loop. Each has a name, each is discussed alone, and the discussion is usually about a condition its dimensions satisfy.

The census explains why. In the plane there is a large set of generic mechanisms to have families of. In space the generic set is nearly empty — one chain at seven links, five at twelve — and everything interesting is a mechanism that moves despite its count, which is a dimensional condition and not a member of any census.

Seventeen links, from the same arithmetic #

The admissible sizes were named as 7, 12, 17 and every fifth number after them, and the census stopped at twelve. The next row is worth taking as far as it goes without a search, because the degree arithmetic alone decides more than it looks as though it should.

At seventeen links the joint count is $(6 \times 17 - 7)/5 = 19$, so the degrees sum to thirty-eight across seventeen links. No link may carry fewer than two, which accounts for thirty-four, and the remaining four are the whole of the freedom an assortment has. Distributing four surplus joints among the links is a partition of four, and there are five: four ternary links; two ternary and one quaternary; two quaternary; one ternary and one five-fold; or a single link carrying six joints.

The same argument reproduces both smaller rows exactly, which is what makes it worth trusting. At seven links the surplus is nought, so there is one assortment and every link is binary — the seven-cycle, arrived at again. At twelve links the surplus is two, the partitions of two are $1+1$ and $2$, and the assortments are ten binary with two ternary, or eleven binary with one quaternary. Those are the two the search found, obtained here without searching.

So the seventeen-link row has five assortments, three independent loops — $19 - 17 + 1$ — and an unknown number of chains. How many of the five contain a mechanism at all is not decidable this way, and the twelve-link row is the standing warning: one of its two assortments is empty, and nothing in the degree arithmetic said so. Settling seventeen means running the same search one size up, on a candidate set several orders larger, and that has not been run here.

What the arithmetic does settle is the shape of the growth, which is the more useful thing. Each admissible size adds five links and six joints, so the surplus rises by two per row, and the number of assortments is the partition count of the surplus:

One, two, five, eleven, twenty-two — the partition numbers, entered by way of a joint count. The planar census reaches two hundred and thirty chains at ten links; the spatial one has five assortments at seventeen, and each of those is a bin that may hold nothing. The difference between the two subjects, argued earlier from a pair of counts, is really a difference in growth rate, and this is where it comes from: in the plane the surplus rises by one for every two links added, and in space by two for every five.

What three loops costs #

The seventeen-link row carries one consequence that twelve does not, and it is worth naming because it is where the field’s own machinery starts to strain.

Twelve links carry two independent loops, and the field’s screw arithmetic handles two loops the way it handles one: a constraint system per loop, stacked, with the rank of the stack read as the mobility. Three loops are not harder in principle and are considerably harder in practice, because the loops share links. A link belonging to all three appears in three constraint systems and its screws must be consistent across them, so the rank of the stack is no longer the sum of the ranks minus an obvious overlap — the overlap is itself a rank computation.

That is the point at which the count and the rank part company for a structural reason rather than a dimensional one. Everywhere in this field so far, the gap between $M$ and the measured mobility has been a condition on lengths and twists: Bennett’s four-bar, Sarrus’s linkage, a planar loop read in space. Here the gap could open because of how the loops are wired, on generic dimensions, and no census of graphs would report it — the graph is admissible, the count is one, and the rank is whatever the shared links leave.

Whether that actually happens at seventeen links is unknown here, and stating it as a risk rather than a finding is the honest form. It is, though, the first place in the spatial census where the two instruments this field runs on could disagree without any special dimension being involved, and it is a reason to want the seventeen-link search rather than merely to note that it was not run.

The joints this census does not have #

One restriction runs through everything above and it is a large one: every joint is revolute.

That is why the arithmetic is so tight. A revolute joint removes five of a body’s six freedoms, which is nearly all of them, so joints and links are almost in balance and the count closes rarely. Give the mechanism other joints and the arithmetic changes completely: a spherical joint removes three, a cylindrical joint four, a prismatic joint five like a revolute but with a different axis structure.

The classical spatial mechanisms make heavy use of the others for precisely this reason. An RSSR loop — revolute, spherical, spherical, revolute — has four links and four joints and counts $6 \times 3 - 3 - 5 - 3 - 3 - 5 = \ldots$ well, it counts to two, one of which is the passive rotation of the coupler about its own axis, which is why the mechanism is described as having one useful freedom and a spinning link.

A census over mixed joint types is a genuinely larger problem: the graph gains a label on every edge, isomorphism becomes isomorphism of labelled graphs, and the admissible link counts multiply rather than thinning out. It is not built here, and the restriction is stated rather than hidden — everything in this essay is about all-revolute chains, which is a smaller subject than spatial mechanisms and is the one whose census is one line of arithmetic away from the planar one.

What is worth taking from a census of one #

There is an obvious objection to a rung whose headline answer is the number one, and it is fair: a census with a single member is not much of a census.

Two answers.

The first is that the one is an explanation. The spatial field’s exclusive use of single loops has looked like a choice of method for as long as the field has existed, and it is not a choice — at seven links there is nothing else. That converts an unexamined habit into a consequence, which is the ordinary business of a census and does not require the census to be large.

The second is that the emptiness is the finding. Six of the nine link counts in the table admit no all-revolute spatial mechanism of one degree of freedom at all, and that is a statement nobody can make from experience: an engineer who has never seen a nine-link spatial linkage has no way of knowing whether that is because there are none or because nobody built one.

Those two together are most of what a small census is for. It is not a design catalogue — five twelve-link chains is not a space anyone needs to search — and it is a complete answer to is that all there is, which is otherwise unanswerable.

What the same routine had to change #

A note about the machinery, since the same enumerator produced both censuses.

Three things are parameterised: the freedoms per body (three or six), the freedoms a joint removes (two or five), and the joint count that a mobility of one implies. Everything else — the search, the pruning rule, the canonical form, the subset scan for structures — is unchanged.

The subset scan needed the same parameterisation and it is worth naming what it now rejects. A spatial subchain of three links held by three revolute joints counts $6 \times 2 - 5 \times 3 = -3$, so a triangle is over-constrained in space rather than merely rigid — which means the spatial census’s rejections are of a different kind from the planar ones, and the rank would see them.

And one thing did not need changing at all: the drawing. A spatial chain’s graph is a graph, drawn by the same layout from the same canonical form, with no geometry anywhere in it — which is one of the few places on this site where the planar and spatial cases need no separate treatment.

And the one thing it settles about the planar case #

A closing observation that runs the other way.

The planar count and the spatial count are the same formula with different constants, and the topology field’s whole apparatus is parameterised on those two numbers. Running it in space is therefore a test of the apparatus as much as a result about space: the same search, the same pruning rule, the same canonical form and the same subset scan, with two integers changed.

That the spatial run produces a sensible and checkable answer — one chain at seven links, forced by an argument short enough to do by hand — is evidence that the planar run is doing what it says. The seven-cycle is the strongest check available, because its answer is derivable without any enumeration at all: every link binary, every binary graph a union of cycles, connected means one cycle, done.