Field

The curve as an equation

Every other field here is handed its demand geometrically: three positions, a sampled path, a ratio. This one is handed a polynomial, and the mechanism is compiled from it rather than searched for — every monomial becomes a cosine, every cosine becomes a link, and the linkage closing is the equation being satisfied. The answer is exact everywhere the machine moves, and what it costs is not accuracy but size.

What the machine draws, against where the polynomial vanishes. Two objects, found two ways. The thin line is the set where x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 is zero, walked over a grid with no mechanism involved. The marks are where the compiled machine's tracing point went, one per converged solve, over the 147 positions of its working arc. The machine's constraint set never mentions the polynomial, so evaluating it at each traced point is an independent check: the worst value over the whole arc is 3.7e-13. The arc is 1.30 radians of the driving angle and not the whole turn, and past that arc it draws something else.

A demand that is an equation

What changes when a mechanism's specification is a polynomial rather than a list of positions: the answer is exact along the whole curve, and what it costs is size rather than accuracy.

Every curve is a sum of cosines

The identity the whole field rests on: a polynomial at the tip of a two-link arm is a finite trigonometric sum with integer frequencies, computed by multiplying Laurent polynomials and checked against a direct evaluation.

What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14.

A circle costs one term

Why the cost of compiling a curve into a linkage is the number of surviving cosine terms rather than the degree, with the dense worst case measured at exactly d² + d.

the reflector, solved. A rhombus whose far vertex is held on a line through the pivot. One side is the input, the line is the mirror, and the other side comes out reflected in it — which is where negation, doubling and addition all come from. The relation it satisfies is (μ, θ) ↦ 2μ − θ, and across a sweep of 41 positions the worst departure from it is 1.3e-13 radians. Every joint here is the output of a Newton–Raphson solve on the bar lengths; nothing is placed by the formula the picture is about.

Four bars that add two angles

The mean and the reflector: two uses of a rhombus that give exact angle arithmetic, measured against their closed forms to 2.7 × 10⁻¹³ across a sweep.

How much of a compiled machine is computing anything. Each machine's bars split two ways: the ones that build an angle — reflectors, means, rigid offsets, the arm — and the ones that carry a direction from where it was computed to where it is needed. On the smallest machines the arithmetic is nearly all of it. By the quintic the carrying is 80 per cent, and it goes on rising, because the arithmetic grows with the number of terms and the carrying grows with the number of pairs of them. That is the answer to why a universality construction is enormous, and it is not about the algebra being hard.

A parallelogram carries an angle, and only so far

The translator: why copying a direction with bars requires a fixed baseline, why the summing chain therefore costs a parallelogram per pair of terms, and why most of a compiled linkage is not computing anything.

What it costs to multiply an angle. Multiplying an angle by n is done by doubling and adding — 2k is one reflector applied to k, and 2k+1 is one addition of k's result and the original — so the cost follows the binary expansion of n and not n itself. Eight costs three doublings; seven costs six gadgets, twice as much, for a smaller number. The other line is what a chain of n−1 additions would cost. The difference matters because the highest multiple a curve of degree d needs is d, so the arithmetic in a compiled machine grows like d log d while the carrying grows like d⁴ — which is why the arithmetic is not what makes these machines large.

Doubling is cheaper than adding

Why an angle multiplier built from reflectors costs about log₂ n rather than n, what the binary expansion has to do with a linkage, and why the arithmetic is the cheap part of a universality construction.

The machine, compiled

The whole pipeline run on one curve: the parts assembled, the machine driven, and the polynomial checked at the traced point by a route the machine's constraints know nothing about.

Every curve in the catalogue, compiled and counted. The bar count is not a formula: it is the number of bar constraints the compiled mechanism actually carries, asked of the object rather than predicted. A line costs five bars and a quintic four hundred and thirteen, which is the field's central number — exactness is paid for in size. The all-revolute column replaces the one prismatic pair that closes the chain with a Peaucellier cell, which costs seven bars whatever the curve, so it is more than half the machine on a line and a rounding error on a quintic. The last column is the polynomial evaluated at the traced point, worst over each machine's working arc.

Five bars for a line, four hundred for a quintic

What compiling a curve into a linkage costs, measured off the mechanisms themselves rather than predicted: bars against terms, terms against degree, and the split between arithmetic and transport.

One linkage, two curves. The same bars, the same lengths, the same driving angle — assembled two ways. One trace is where the polynomial vanishes and the other is not: the worst value of xy − 0.5 along the second is 8.7e-1, against 4.7e-14 along the first. Every position on both was solved to 9.5e-14. Nothing about the second linkage is defective; one of its parallelograms is a crossed one, so a direction is being carried wrongly, and the machine is faithfully computing a different function.

The proof drew more than the curve

The gap in Kempe's argument, measured: a parallelogram's second assembly makes a compiled machine compute a different function, and every constraint it has is still satisfied exactly.

A brace is one redundant equation, on purpose. The compiled machine, counted and measured, with and without 4 braces. The count says the braced machine has -3 degrees of freedom — it cannot move — and the rank of the constraint Jacobian says it has 1, the same as before. Every brace contributes exactly one equation the others already imply, which is what overconstraint is, and here it is being added deliberately: the redundancy is what removes the assemblies the count knows nothing about. This is Grübler being wrong for the useful reason rather than the embarrassing one.

A bar between two midpoints

The brace that removes a parallelogram's second assembly, what it costs in redundant equations, and why the count says the braced machine cannot move while the rank says it moves exactly as before.

What a singularity does, and what it does not do. The reflector driven straight through the configuration at which its two placements merge — here θ = 0.800, where the rhombus flattens onto its own mirror. Two numbers are plotted. The closure residual is how well the bars are satisfied, and it does not move: 8.3e-14 on both sides. The departure is how far the output is from the angle the gadget is supposed to produce, and it goes from the floor to order one at 0.800. Nothing breaks. The gadget goes on being a perfectly good linkage and stops being the function it was built to be.

Where the machine stops being the function

The singular configurations of the gadgets, why they bound a compiled machine's working arc, and why no bracing removes them.

Exactness is not bought with links. Five straight-line mechanisms, each measured over its own working arc — walked out to its dead centres and back a tenth — and each plotted at its own bar count. Watt's four bars are wrong by 9.0 per cent of the stroke and Chebyshev's by 12.4; Peaucellier's seven are exact. There is nothing in between, and adding bars to an approximation does not walk down the axis: the compiled machine is exact for the same reason Peaucellier is — an exact algebraic relation — and its extra bars buy generality rather than accuracy.

Exact costs more than close

Five straight-line mechanisms measured over their own working arcs: two approximations at nine and twelve per cent, three exact ones at the solver's floor, and nothing between them.

Bars against terms, over the whole catalogue. One mark per compiled machine. The bar count rises much faster than the term count, and the reason is the summing chain: term k has to have its direction carried to the k−1th vertex of the chain, one parallelogram per hop, so the carrying costs a translator for every pair of terms. Nine curves, from five bars to four hundred and thirteen, on a term count that goes from one to eighteen.

What universality is worth

What a universality construction gives and what it does not: the honest accounting after nine machines were compiled, driven and measured.

A compiled machine and its own scale

What a scaling does to a machine compiled from an equation, why its curve is preserved and its function is not, and what that says about identifying one.

A rectangular hyperbola, compiled from a multiple of its equation. The machine compiled from p · (1 + x² + y²) for a rectangular hyperbola, with the translators — the parallelograms that carry a direction from where it is produced to where it is needed — in their own colour. The factor 1 + x² + y² is at least one at every real point, so every equation here vanishes on exactly the same curve. The machines do not agree: 20 bars at p, 75 bars at p · (1 + x² + y²), 144 bars at p · (1 + x² + y²)². This one solves 29 positions over an arc of 0.508 radians, and at every one of them the original polynomial reads 4.93e-14.

The price is on the equation

One curve compiled from three equations of it: bar counts of 20, 75 and 144 for a hyperbola whose real points never move, the original polynomial still reading 5 × 10⁻¹⁴ at the padded machine's own positions, a penalty factor running from 2.07 to 10 across four curves, and the working arc falling to nothing.

Two circles, one term. The curve r² = 1.44 together with r² = 2.56, whose squared radii sum to 4.00 — four times the square of the arm's link length. Their product equation expands to 1 term, which is what either circle costs on its own, so the second circle is free. The machine compiled from it has 17 bars against 11 for the single circle, runs over 5.200 radians against 1.560, and stays on the outer component throughout: its radius varies by 4.15e-13 over 240 solved positions. A mechanism moves continuously and the two circles are disjoint, so no assembly of it reaches both.

Two circles for the price of one

An exact sparsest-vector search over the multiples of four catalogue curves, 165 to 2300 solves each, finding the curve's own equation minimal every time; and the one tie, which adds a whole second circle for nothing and generalises to six circles at three terms by a parity argument on cos θ.

The cheapest curve of each degree. The curve of degree 6 whose equation expands into the fewest cosines of the arm's two angles: cos 3θ = 0.3: 3 concentric circles about the pivot. It costs 1 term — cos(3α − 3β + …) — and the machine compiled from it has 27 bars. The dashed circle is the arm's reach. The dots are the machine's tracing point over 140 solved positions; it draws one component and stays on it. At every even degree the cheapest curve costs one term and at every odd degree two, whatever the degree.

The cheapest curves at each degree

An exhaustive search over the whole space of polynomials of degree one to six for the sparsest expansion in cosines of the arm's two angles: the cheapest curve at each degree with its compiled machine; the census table of supports tried and the one support reaching the minimum; the frequency pairs and their mirrors under swapping the links; k concentric circles at cos kθ = c for one term; the odd-degree curves, a line through the pivot times circles; the price list for conics, with nothing at two terms; and the bars each cheapest machine costs.

A quintic of eighteen terms, and the nearest curve that costs a few. The quintic x^5 + 0.4x^2y − y = 0, whose equation expands into 18 cosines of the arm's two angles, inside the arm's reach (dashed). Drawn over it, the curve of the same degree with at most 4 terms that comes nearest to it inside the reach, found by trying every set of terms of that size: it keeps 0,1|1,0, 1,-2|2,-1 and is at most 2.77e-2 from the quintic and the quintic at most that from it — 1.4% of the reach. The slider steps through the budgets at which the nearest curve improves.

Four terms of eighteen draw the quintic

Six catalogue curves priced by the nearest curve of the same degree with a budget of terms, searched over every set of mirror classes: the quintic and its nearest four-term curve inside the arm's reach; the price curves of all six; the quintic's classes by amplitude and the classes each nearest curve keeps; searched against the largest terms refitted; the folium and its nearest ten-term curve; and each curve's degree, terms and price at half.

Moving the pivot adds terms everywhere but one locus. The folium x³ + y³ − 1.5xy = 0, drawn in its own coordinates, and the square of pivot positions from −1 to 1 searched at steps of 0.05 (shaded). With the arm's pivot at the origin, where the catalogue puts it, the equation expands into 11 cosines of the arm's two angles. Moved anywhere in the square off the line x + y = 0 (heavy), it expands into 12; at (0.5, 0.3), for instance, 12. 41 of the 1681 pivots tried cost less than 12, all on that locus, and none costs less than the catalogue's 11.

Moving the pivot only adds terms

Eight catalogue curves with the arm's pivot moved over a square of 1,681 positions: a curve and the one locus of pivots where its equation is cheapest; every curve's count at the catalogue's pivot, at its least and elsewhere; the folium's one cancelling term across its line; the quintic's count along two lines through its pivot; the ellipse's nearest three- and four-term curves from every pivot that reaches it; and the cosines a move adds to the ellipse.

Along one line the nearly free curve stays nearly free. The arm's pivot is fixed at the centre of its reach (dashed); the quintic x⁵ + 0.4x²y − y = 0 is drawn with its own point — where the catalogue put the pivot — moved 0.2 of a link along the line x = 0, which is the same as moving the pivot the other way. Heavy: the arc priced, the part of the quintic within 1.4 of its own point. Thin: the nearest curve built from at most four cosines of the arm's angles, found by trying every set of four: 6.0% of the reach away, the line through the pivot bent by the square of the radius. The equation itself costs 22 cosines here.

The quintic's cheapness runs along a line

The catalogue's quintic priced from moved pivots: the arm fixed with the quintic moved and its nearest four-term curve; the shorter arc that every pivot reaches; the four- and six-term prices along the line x = 0 and across it; the amplitude each move adds to the equation's cosines; the cosines of the moved equations side by side; and every pivot's exact count and prices as a table.

The quintic's cheap pivots lie off the line

The catalogue's quintic priced from a grid of pivots: the four-term price over the grid, with its five per cent region, its valley and the line x = 0; the quintic and its four-term curve from pivots in the valley and on the line; the price across at four heights; the price along the valley and along the line; the census at two valley pivots; and the valley height by height.

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