Six things a compiled linkage is not

Six things that get said about linkages built from an equation. None of them is careless, every one is exactly right about something, and every one is answered here with a number from the branch census, the cost table or the working arc.

Two of the six are about mistaking a measurement for a verdict, two about mistaking a theorem for a design, and two about mistaking one quantity for another that happens to have the same units of importance.

Nought: the list itself #

One preliminary, because this is the seventeenth of these lists on this site and the form has a hazard of its own.

Every claim below is one somebody could reasonably hold after reading the field’s own opening rung, and none is a straw position. They are the readings a careful person arrives at from a correct account, which is what makes them worth answering: a misconception nobody would form is a misconception nobody needs correcting.

They are also, all six, statements the work behind this field held at some point while it was being built. The closure residual really was the only thing being watched until the branch census was run. The cost really was expected to follow the degree until the lemniscate came out at fifty bars and the cubic at a hundred and one. A list of things the work got wrong is a more honest object than a list of things a reader might, and this one is the former.

One: a closed loop means a correct mechanism #

It means the bars are the lengths they were made. That is a great deal — it is the site’s founding invariant, and a figure showing a configuration the mechanism cannot reach is the failure that matters almost everywhere — and it is not a statement about what the mechanism computes.

The machine compiled from $xy = 0.5$ has four parallelograms, so sixteen ways to be put together. Eight of them close. Four put the tracing point on the curve; four put it on a different curve entirely, with the polynomial reading $0.15$ where zero was wanted.

All eight close to $9.6\times10^{-15}$.

There is no threshold that admits the right four and rejects the wrong four, because the two groups are not separated in that quantity. The only thing that separates them is a measurement taken from outside the constraint set — the polynomial, evaluated at a point the mechanism reached without knowing the polynomial exists.

This is the general form of a trap the site keeps meeting from different directions. A Grübler count says a working mechanism cannot move. A screw system’s rank says two mechanisms are alike when they are not. Each time, the instrument is measuring something real and being read as measuring something else.

Two: the theorem says any curve can be drawn, so the design problem is solved #

The theorem says a linkage exists. It is silent about how big and about how much of the curve.

Built and measured: a general quintic costs four hundred and thirteen bars, of which three hundred and eight do no arithmetic at all, and it draws its curve over 0.17 radians of driving angle — about ten degrees. Braced, so that it has no assemblies drawing other curves, it has one thousand and ninety-six unknowns rather than four hundred and eighty.

A solved design problem would have to say something about both of those numbers. The theorem says nothing about either, and that is not an oversight: a statement of the form there exists is silent about size by construction, and the bounds proved since are enormous.

The right reading is that the question can a linkage draw this curve has been closed, for every algebraic curve at once, and the question what is the smallest linkage that draws it has not been opened. The compiler’s answer is an upper bound, and on the one case where a better answer is known — the sextic an ordinary four-bar traces — the bound is a hundred times too big.

Three: exact means accurate #

Exact means an identity holds. Accurate means an artefact does what its drawing says, and the second is a property of parts rather than of relations.

Lengthen one bar of a compiled machine by $10^{-4}$ and the tracing point leaves the curve. How far depends on which bar and on how large the machine is: on the smallest machine the polynomial reads $8.6\times10^{-5}$, less than the error put in, and on the lemniscate’s fifty-bar machine it reaches $9.1\times10^{-3}$ — an amplification of ninety.

So an exact construction whose errors are amplified ninety times is not, in any sense a builder would recognise, more accurate than an approximation whose errors are not amplified at all. The practice field has said this about every exact mechanism on this site; what is new here is the size at which it bites, and that the amplification grows with the construction. A bigger proof is a worse artefact.

Three and a half: which bars matter is the ones that cost most #

An immediate corollary of the third claim, and it comes out the opposite way round from every expectation the cost table sets up.

The bars of a compiled machine split into arithmetic — reflectors, means, rigid offsets — and transport, and the transport is three quarters of the largest machine. So the natural guess is that the transport is where a made machine’s errors come from, because there is so much of it.

It is not. Lengthening bars one at a time and reading the polynomial at the tracing point, the worst offenders are consistently reflectors, and the translators barely move the answer at all.

The reason is that a reflector is doing arithmetic on an angle and a translator is copying one. An error in a reflector’s rhombus changes an angle that is then multiplied up and combined; an error in a parallelogram shifts a copied direction by a first-order amount and nothing compounds it.

Size and fragility live in different parts of the same machine, which is a bad property for anything anybody wanted to build: the part to make carefully is not the part there is most of, and it is not the part a builder’s attention would naturally go to.

Four: a higher-degree curve costs more #

Degree is not the cost. The number of surviving cosine terms is, and cancellation depends on the coefficients.

A lemniscate is degree four and expands to five terms: fifty bars. A general cubic is degree three and expands to eight: a hundred and one bars. The lower-degree curve costs twice as much, because the lemniscate is symmetric about both axes and about the origin and the cubic is symmetric about nothing.

Nor does the monomial count help. A rectangular hyperbola is written with one monomial and expands to three terms; a lemniscate is written with five and expands to five, and not by any correspondence between them. The two counts measure different things: one is how the curve was written down, the other is how the arm’s angles enter it.

The dense worst case is $d^2 + d$ terms at degree $d$, measured exactly at six degrees. Every real curve sits under that ceiling, and how far under is the amount of structure it has.

Four and a half: a sparse polynomial gives a small machine #

The most reasonable-sounding of all of these, and it fails on the smallest example in the catalogue.

$xy - 0.5$ is one monomial and a constant. Its machine is twenty bars. $x^2 + y^2 - 1.44$ is three monomials and its machine is eleven.

A single monomial spreads. $xy$ is a product of two four-term Laurent polynomials, so sixteen products go in and three cosines come out, at frequencies $(2,0)$, $(0,2)$ and $(1,-1)$. Writing the curve compactly did nothing to make the machine small, because compactness in $x$ and $y$ is not compactness in $\alpha$ and $\beta$.

The circle is small for the opposite reason: its three monomials produce sixteen products of which fourteen cancel, leaving the cosine rule. Cancellation is what makes a machine small, and cancellation is a property of the coefficients, which is exactly the thing a monomial count throws away.

Five: the compiler searches for a mechanism #

Nothing is searched for. There is no starting guess, no objective function, no root count and no candidate to reject.

The polynomial is multiplied out; the surviving frequency pairs are read off; each becomes a stack of gadgets whose composition is fixed by the binary expansion of its two integers; the results are summed by a chain whose length is the number of terms. Every bar length is determined by the input, and there is nothing left to choose.

That is what makes the field a neighbour of synthesis rather than a part of it. Synthesis takes positions and returns a finite set of candidates, most of which turn out to be defective and have to be tested. Compilation takes an equation and returns one machine, in under a second, with nothing to test afterwards except that it does what the procedure says.

The price of not searching is that nothing is ever recognised. Handed a circle, the compiler produces eleven bars where a crank would do — and the eleven bars are a crank, wrapped in apparatus that had nothing to sum. A procedure that never recognises anything is what general means.

Five and a half: the arithmetic is what makes it big #

Not one of the six, because nobody says it out loud, but everybody assumes it, and it is worth a paragraph because it is the finding this field would not have got without building something.

Reading the construction, the interest is all in the angle gadgets — the reflectors, the doublings, the binary expansion of an integer showing up in a mechanism. The parallelograms that carry directions around are the boring part.

Counting them says the boring part is three quarters of the largest machine, and the fraction is still rising. The arithmetic grows with the number of distinct multiples, which is about $2d\log d$; the carrying grows with the number of pairs of terms, which is about $d^4$.

So anybody setting out to make a smaller universal construction would naturally attack the arithmetic, and would be optimising the quarter that grows slowest. The place to attack is transport, and transport is not a feature of this construction: it is a property of what a bar can say, which is that two points are a fixed distance apart, and nothing else.

Six: the machine draws the curve #

It draws the curve on an arc, and past the arc it draws something else while continuing to satisfy every constraint it has.

The arcs measured here run from $2.0$ radians on a line down to $0.17$ on a quintic. What ends one is a gadget reaching a configuration at which two of its placements merge — a rhombus flattening onto its own mirror — and a continuation through that point may come out on either. Nothing breaks. The closure residual is $8.3\times10^{-14}$ on both sides.

Bracing does not help, and the reason is worth keeping separate from the reason bracing helps with branches. A branch is two configurations at a finite distance, one of which is intended, and a constraint true in one and false in the other removes it. A singularity is where two configurations coincide, and no constraint distinguishes them there because at that instant they are the same configuration.

The classical statements of the theorem are careful about this: the linkage traces the curve in a neighbourhood. What this field adds is how big the neighbourhood is, which the theorem does not say and which turns out to be measured in degrees.

A seventh, which is about this site rather than about linkages #

There is one more claim worth answering, and it is one a reader of the earlier fields might make against this one.

A machine of four hundred bars cannot be checked, so its numbers cannot be trusted.

The first half is true. Nobody can look at a four-hundred-bar linkage and see whether it is right, and no figure in this field pretends otherwise — the scenes are painted by gadget kind precisely because following one joint by joint is not possible.

The second half does not follow, and the reason is the discipline the whole collection runs on. Three independent measurements are taken of every machine. The closure says the bars are satisfied. The curve residual evaluates the polynomial at the tracing point, which the machine’s constraint set never mentions. The departure compares each gadget’s actual output angle against the angle its own specification asks for.

Those three can be wrong together only if the compiler, the solver and the polynomial evaluator are wrong in exactly compensating ways, and they share no code. On top of that the identity behind the whole construction was checked against a direct evaluation at twenty-seven hundred random angle pairs, and each gadget was measured separately on its own bench before any of them were assembled.

A large object is checkable when the checks are independent of it. That is the same argument the topology field makes about a census of three thousand candidate graphs, and the algebra field about a root count nobody can verify by hand.

What the six have in common #

Four of them are the same mistake in different clothes: a quantity that is correct is read as an answer to a question it was not asked. The closure residual is right about the bars and silent about the function. The theorem is right about existence and silent about size. Exactness is right about the relation and silent about the artefact. The degree is right about the frequencies available and silent about which survive.

The other two are about generality being expensive, which is the field’s own subject. A procedure that never searches cannot recognise, and a construction that works everywhere works on a neighbourhood.

There is a reason all six show up in this particular field and not in the others, and it is worth naming. In every earlier field on this site the mechanism’s output is a shape or a motion — where a point goes, whether a link turns, how far a follower rises — and a mechanism that is wrong about its output is visibly wrong. Here the output is the value of a polynomial at a point, and nothing observes it. A compiled machine on a spurious branch runs smoothly, looks entirely normal, and draws a plausible curve.

A mechanism whose output is a computation hides its own failures, and every one of the six is a version of that. It is also the reason this field needed three instruments where the rest of the site has managed with one.

None of the six is answerable by being more careful with the instrument at hand. Each one needs a second measurement, taken from somewhere the first cannot reach — which is the habit this whole collection is built on, and the reason a compiled machine is checked by a polynomial its own constraints have never heard of.

The six share a shape and it is the one this site keeps meeting from new directions: a property that holds of an object is read as a property of the thing that produced it, or the reverse. A closure residual is a property of a solve and is read as a verdict on a mechanism. A theorem is a property of a class and is read as a design for a member. Exactness is a property of a construction and is read as accuracy of a part. A degree is a property of a curve and is read as a cost of a machine. In every case the correct statement is about one level and the wrong one is about the level next door, and the two are close enough that the substitution is invisible. The remedy is unglamorous and it is the same one the site applies elsewhere: say which object each number is about, in the sentence that quotes it. A residual belongs to a solve at a configuration; a bar count belongs to a compiled machine for a stated polynomial; an exactness belongs to an ideal geometry. Attaching the object costs a clause and it is the clause every one of the six is missing.