Five bars for a line, four hundred for a quintic

The construction works. This rung asks what it costs, and the answer is the field’s central number.

The table, and how the numbers were got #

Nothing in that table is a cost model. Each row is a Mechanism that was built, assembled, driven and measured; the bar count is the length of its constraint list, the joint count is the length of its joint list, and the last column is the polynomial evaluated at the tracing point over the whole of the machine’s working arc.

That distinction has already earned its keep. An early version of the compiler produced machines whose bars were all present and whose Jacobian was rank-deficient by eight on the cubic, because some of those bars had come out collinear. A formula would have reported the same bar count either way. What found it was asking the machine to move.

What the numbers say #

A line costs five bars. Two terms, one small chain, and a slide. Fewer bars than Peaucellier’s cell, which is seven — though the comparison is not fair yet, and the section on the closure below is where it becomes fair.

A circle costs eleven, and a crank costs one. The expansion gives a circle a single term, and the machine built around that single term is an arm with a fixed elbow angle — which is a crank, wrapped in ten bars of apparatus that had nothing to add up.

A rectangular hyperbola costs twenty. A parabola costs fifty-two. Both are conics. The parabola is expensive because it mixes degrees and mixing degrees prevents cancellation.

A lemniscate, degree four, costs fifty. A general cubic, degree three, costs a hundred and one. The lower-degree curve costs twice as much, because it has no symmetry and the lemniscate has two.

A general quintic costs four hundred and thirteen bars and two hundred and forty-two joints.

Bars against terms #

The term count is the quantity that governs the size, and the relation between them is not linear.

Eighteen terms is eighteen times as many as one, and four hundred and thirteen bars is thirty-seven times as many as eleven. The extra factor is the summing chain: each term needs its direction carried to the chain vertex before it, one parallelogram per hop, so the carrying costs a parallelogram for every pair of terms. That is $N(N-1)/2$, and for eighteen terms it is a hundred and fifty-three.

Put the two growth laws together. A dense curve of degree $d$ has $d^2 + d$ terms, so the carrying goes as the square of that, which is $d^4$. A universality construction’s size is a fourth power of the degree, and the fourth power comes from transport.

The split, which is the finding #

On the line the carrying is two bars of five. On the hyperbola, eight of twenty. On the lemniscate, twenty-two of fifty. On the quintic, three hundred and eight of four hundred and thirteen — three quarters of the machine.

And the fraction is still rising, because the arithmetic grows with the number of distinct multiples and the carrying grows with the number of pairs of terms.

That is the answer to why is a universality construction so enormous, and it is not the answer the question invites. The invited answer is that the algebra of a high-degree curve is complicated. The algebra is a hundred bars. What is expensive is that a linkage has no way to refer to a quantity computed somewhere else — no variables, no names, no wires — and every reference has to be built out of matter.

Reading the growth honestly #

A nine-point curve fitted with a fourth power is a claim that deserves a caution, because nine points can be fitted with a great many things.

The $d^4$ is not a fit. It is arithmetic: the dense term count is $d^2 + d$, measured exactly at six degrees; the transport is $N(N-1)/2$ parallelograms, which is a count of what the compiler builds rather than an observation about it. Composing the two gives a fourth power with no data involved.

What the nine machines demonstrate is a different and weaker thing: that the arithmetic is not an over-estimate, that the machines really do get that large, and that nothing in the implementation is quietly avoiding the cost. They also show the enormous spread around it. The lemniscate at degree four costs fifty bars where the dense law would say something near four hundred, and the whole gap is symmetry.

So the fourth power is the ceiling and the curves live under it, at distances that have nothing to do with degree. Quoting it as the growth would be quoting the worst case as the typical one, which on this catalogue would be wrong by an order of magnitude on three rows out of nine.

The closure, and the fair comparison #

One column of the table has not been used yet, and it settles the comparison with Peaucellier.

The compiled machine’s last constraint holds the chain’s end on a vertical line, and a slide does that in one prismatic pair. Kempe would not have allowed it: the whole interest of his result was that a linkage of revolutes could do this, and a slider is a different lower pair.

Replacing the slide with a Peaucellier cell — whose output point runs on an exact vertical line — costs seven bars and five joints, measured, on every curve in the catalogue. The cell’s size has nothing to do with the polynomial, so it is a constant.

That constant lands very differently at the two ends of the table. On the line it takes the machine from five bars to twelve, so the compiler’s answer for a straight line is Peaucellier’s cell plus five bars of overhead — which is exactly the right result for a general procedure applied to a case that has a famous special answer. On the quintic it takes four hundred and thirteen to four hundred and twenty, which is under two per cent.

The counts quoted elsewhere in this field use the slide, because that keeps them about the compilation rather than about the closure. The all-revolute figure is beside it in the table.

Two counts that are not the bar count #

The table carries joints as well as bars, and the two do not move together in the way a reader of the rest of this site will expect.

For an ordinary planar linkage the counts are locked to one another: a chain of $n$ binary links has about $n$ pins, and Grübler’s formula reads both. Here the ratio drifts. The line has five bars and ten joints — twice as many joints as bars — and the quintic has four hundred and thirteen bars and two hundred and forty-two joints, which is the other way round.

The reason is the rigid offsets. A triangular link is one joint and one attachment constraint, and it carries no bar at all; a parallelogram is one joint and two bars. So a machine dominated by offsets counts more joints than bars, and a machine dominated by transport counts more bars than joints. The crossover happens between the circle and the hyperbola, which is to say almost immediately.

That is worth stating because it makes the joint count the wrong quantity to quote. A bar is a part — something to be made to a length — and a rigid offset is one part carrying three points. Counting bars counts things a machinist would make; counting joints counts places where two of them meet, and the two are not proportional in this construction the way they are everywhere else on this site.

The arc, which does not follow the size #

There is a fourth column in the machinery that the table reports and nothing above has explained: how far each machine turns.

The working arcs run from $2.0$ radians on the line and the folium down to $0.17$ radians on the quintic. That is about ten degrees of driving angle, on the largest machine here.

The relation to size is real but it is not monotone. The folium, at a hundred and sixty bars, works over $2.0$ radians; the lemniscate, at fifty, over $1.3$. What ends an arc is the first gadget to reach a singular configuration, and a machine with more gadgets has more chances to reach one — but when it does depends on where its term angles happen to be, which is a fact about the curve rather than about the count.

So a bigger machine is not simply a machine that works less, and the two costs are separate. A compiled quintic is four hundred bars that draw ten degrees of a curve, and putting those two numbers in one sentence is the honest summary of what this construction gives.

What is not in the count #

Three costs the table does not carry, each of which a builder would meet immediately.

The braces. Every parallelogram has a second assembly, and removing them costs a bar and two rigid attachments each. On the quintic that is a hundred and fifty-four braces, taking the machine from four hundred and eighty unknowns to one thousand and ninety-six. The rung on bracing has the arithmetic; here it is enough to say that the braced machine is more than twice the unbraced one, and that the counts in this table are for the unbraced version.

The link lengths. The amplitudes within one curve’s expansion span a factor of a hundred on the quintic — $0.063$ to $6.3$. Every one of those is a bar that has to be made, and the small ones have to be made to a hundred times the relative precision of the large ones for the same absolute error.

The thickness. Nothing here has any. Four hundred bars in a plane, many of them overlapping, is not an assembly anybody can build without lifting them into layers, and the layers are a mechanical problem this site does not model. It is worth naming as a real omission rather than leaving as an implication of the drawings.

Against the historical figures #

The counts above are counts of this construction, and the difference from the historical ones is worth setting out because the numbers usually quoted for Kempe’s linkages are much larger.

Kempe’s own gadgets are contra-parallelogram based and his multiplicator and additor are more expensive than the rhombus pair used here. His architecture is the same — arithmetic at a fixed pivot, a summing chain, a line closure — so the fourth-power growth in the transport is his too, and it is the dominant term in both constructions. What differs is the constant in front of it and the cost of the arithmetic, and both are larger for him.

There is also a long history of improvement. Later work brought the counts down substantially and gave a proper account of the degenerate assemblies Kempe’s original argument passed over, which is the subject of the next rung. None of that arithmetic is reproduced here.

So no number in this field should be read as a statement about Kempe’s linkage. They are statements about a construction built to be positioned by this site’s solver and checked by this site’s gates, and where a comparison with the historical figures appears it is qualitative and says so. What the two have in common is the shape of the cost, and that is the part worth carrying anywhere else.

Why measure a construction nobody would build #

A fair question, and the answer is the reason this field is a field.

Because the count is the theorem’s content. Every algebraic curve is drawn exactly by some linkage is a statement with no numbers in it, and a statement with no numbers cannot be wrong in an interesting way. The version with numbers — a quintic costs four hundred and thirteen bars, over an arc of ten degrees, three quarters of it transport — says something a reader can act on: that universality is real, that it is useless, and precisely which part of it is useless.

Because the shape of the cost is transferable. The observation that a linkage pays for every reference to a distant quantity is not about Kempe. It applies to any construction built from bars, and it says where to look for a better one. Somebody who wanted a smaller universal construction should not be optimising the angle arithmetic, which is a quarter of the machine and grows slowest; they should be attacking transport, and they will find that transport is not a shortcoming of a method but a property of what a bar can say.

Because the counts are checkable. Every number here comes off an object that was built and driven, and every one of those objects put its tracing point on its own curve to within $10^{-11}$. A cost model that agreed with a machine nobody built would be a cost model about nothing.

Where the cost stops mattering #

The last thing to say about the size is that it is not the reason these machines are unused, and the frontier rung is where that lands.

A compiled straight-line machine has five bars and is exact to $10^{-14}$ of its stroke. Watt’s linkage has four and is wrong by nine per cent. On any account of engineering that weighs exactness against parts, the compiled machine wins that comparison outright, and it is not used.

So the size is not the objection, or not the whole one. What actually decides it is what a made machine does to an exact relation — where the answer is that the amplification of a length error grows with the machine, reaching ninety on the lemniscate — and the fact that nine per cent of a stroke is very often good enough while four hundred bars is never convenient.

That is the shape of the whole subject, restated at this field’s scale. Watt’s linkage is in every beam engine ever built and Peaucellier’s cell is in a museum, and the reason has never been that anybody doubted the arithmetic.

One number to leave with #

If a reader carries one figure from this rung it should not be four hundred and thirteen. It should be three quarters.

Three quarters of the largest machine here does no arithmetic. It computes nothing, decides nothing, and would be unnecessary if a linkage had any way of saying the angle over there. Every parallelogram in that three quarters is a perfectly good mechanism doing a perfectly trivial job, and there are three hundred and eight of them.

That is a statement about linkages rather than about this construction, and it is the piece most likely to be useful somewhere else. A strand carries a length along a path without a bar per unit of path, which is why the strand field exists; nothing in the subject carries a direction that way. Until something does, any construction that computes in one place and uses in another will spend most of itself on the journey.

Three quarters of the largest machine doing no computing is the finding, and its general form is worth stating because it is a fact about compiled constructions rather than about linkages. A computation laid out in space pays for geography. Every intermediate result has a place, every place is a distance from where the result is needed, and moving a quantity costs hardware in proportion to that distance. So a construction’s size is not the count of its operations; it is the count of its operations plus the cost of getting each result to where the next one is. That is a cost with no analogue in a computation done in registers, where a value is available everywhere at once for nothing, and it is why the growth here is a fourth power of the degree rather than the second power the term count alone would give. The lesson transfers to anything built rather than executed — a mechanical computer, a fluidic circuit, a network of linkages — and it says the same thing in each: the layout is most of the machine, and an accounting that counts only the operations will understate a design by whatever the transport costs.