Four bars that add two angles

A compiled machine has to produce a link at angle $m\alpha + n\beta + \varphi$ for every term of the expansion. Two angles go in, whole-number multiples of them come out, and everything in between has to be done with bars.

Two constructions do all of the arithmetic. Both are a rhombus, used two different ways.

The mean, which is a rhombus and nothing else #

Take a pivot $O$ and two links from it of the same length $s$, at angles $\theta_1$ and $\theta_2$, ending at points $A$ and $B$. Add two more bars of the same length, from $A$ and from $B$, meeting at a point $D$.

The four bars form a rhombus $O$, $A$, $D$, $B$, and in a rhombus the far vertex is the vector sum of the two sides:

$$D - O = (A - O) + (B - O).$$

Both sides have length $s$, so their sum lies along the bisector. The direction of $OD$ is exactly $(\theta_1 + \theta_2)/2$, and there is nothing approximate in the statement: it follows from two vectors of equal length adding along their bisector, which is the reason a rhombus’s diagonals bisect its angles.

Two bars, one new joint, no freedom added or taken away: $D$ has two coordinates and two bar constraints, so the gadget is determinate — given its inputs, its output is fixed.

The one thing to notice about it is what happens at $\theta_1 - \theta_2 = \pi$. The two vectors are opposite, their sum is zero, and $D$ arrives at the pivot with no direction at all. That is the mean’s singular configuration and it is a real place, not an artefact; the rung on where the machine stops working is about what happens when a machine drives through one.

The reflector, which is the same rhombus with a rail #

Now do something different with the same four bars: instead of taking the diagonal as an output, force it to lie along a given line and read the fourth vertex.

Give the pivot a mirror direction $\mu$ — a line through $O$, materialised as a point on it — and one input link at $\theta$. Build the rhombus so its far vertex $D$ lies on that line. Then the other side comes out at

$$2\mu - \theta,$$

which is the input reflected in the mirror.

The bookkeeping: $D$ and $B$ are new, four unknowns; three bars and one condition holding $D$ on the line, four equations. Determinate again, and one gadget more expensive than the mean.

Everything from the reflector #

The reflector is a two-input, one-output device and it is enough on its own.

Negation. Reflect the input in the frame direction. $2 \cdot 0 - \theta = -\theta$.

Doubling. Reflect the frame direction in the input. $2\theta - 0 = 2\theta$. The roles of mirror and input swap, and the same four bars that negated now double.

Addition. Take the mean of $\theta_1$ and $\theta_2$ with a free rhombus, then reflect the frame direction in that: $2\cdot\frac{\theta_1+\theta_2}{2} - 0 = \theta_1 + \theta_2$. Two gadgets, six bars, three new joints, and two angles have been added.

Subtraction is addition and a negation. Multiplication by a whole number is repeated doubling and addition, and which combination is cheapest turns out to depend on the number’s binary expansion rather than on its size.

That is the whole arithmetic. Two constructions, four operations, and every integer combination of two angles reachable from them.

The reflector’s two solutions, written out #

The reflector is the one gadget whose branch structure can be written down in closed form, and doing so explains both of its singularities and most of what goes wrong later.

Put the pivot at the origin, the mirror along $\mu$, the input link at $\theta$ with length $s$. The far vertex $D$ has to be on the mirror, so $D = d\,e^{i\mu}$ for some scalar $d$, and it has to be at distance $s$ from the input’s end. Squaring that condition:

$$d^2 - 2ds\cos(\mu - \theta) = 0,$$

so $d = 0$ or $d = 2s\cos(\mu - \theta)$.

The first root is the degenerate one: the whole rhombus collapsed to the pivot. It satisfies both bars and it is not a configuration anything can pass through, since the two remaining bars would then be free to point anywhere. The second is the working one, and its length varies with the angle between the mirror and the input.

Two consequences fall straight out.

At $\mu - \theta = \pm\pi/2$ the working root arrives at zero and merges with the degenerate one. That is one of the reflector’s singularities and the rhombus is closed up on itself there.

At $\mu = \theta$ the vertex is at its furthest, $d = 2s$, and the rhombus is a straight line: the input, the vertex and the output all along the mirror. The two placements of the output merge there, because the two circles that place it are tangent. That is the other singularity, and it is the one a compiled machine actually hits, because the doubler reflects the frame direction in $\alpha$ and $\alpha$ passes through zero on any decent working arc.

That second one is worth naming precisely, because a reader watching a machine leave its curve later will want to know what it did. It did not break a bar or fail to converge. It reached a configuration where two solutions of a quadratic coincided, and came out on the other one.

The vertex changes sign, and that is not a failure. For $|\mu - \theta|$ greater than a quarter turn, $d$ is negative and the vertex sits on the far ray of the mirror line. The gadget goes on working perfectly — a mirror is a line, and reflecting in a line does not care which way along it anything points. Any instrument that compared the vertex’s direction against the mirror’s would report a departure of $\pi$ at every such crossing and would be wrong about a machine that was right, which is why the vertices whose meaning is a line are compared modulo $\pi$ and the output links, which have a direction, are not.

The phase, which is free #

The expansion’s third number is a phase, and it needs no gadget at all.

A point attached rigidly to a link — at a fixed angle to it and a fixed fraction of its length — is at the link’s direction plus a constant, at a length the attachment chooses. Physically that is a triangular link: one rigid piece with three points on it. Given a link from the pivot at angle $\theta$ and length $s$, one such attachment produces a point at angle $\theta + \varphi$ and radius $R$, for any $\varphi$ and any $R$.

So a term’s phase and its amplitude — two of its three numbers — cost one triangular link between them, and the frequencies cost everything else. That asymmetry is worth carrying: a compiled machine’s size is governed entirely by the integers in the expansion, and not at all by the real numbers.

The offset has one more property that separates it from the other three gadgets, and it matters later. A rigid attachment has one placement. Two bars meeting at a joint have two — the two intersections of two circles — and every gadget built from bars therefore has a second assembly. The offset does not, because a rigid body attached to a rigid body is where it is.

What the four are worth, measured #

Each gadget was built as a mechanism on its own bench, driven across a sweep, and its output compared against the closed form it is supposed to satisfy.

Rigid offset $2.2\times10^{-16}$, mean $2.7\times10^{-13}$, reflector $1.6\times10^{-13}$, translator $1.7\times10^{-13}$. All four at the floor.

Two things about that measurement are load-bearing.

Each configuration is solved from the previous one, not from the closed form. Seeding a Newton solve with the answer and then observing that it agrees with the answer is a check that tested nothing, and this site has made that mistake and recorded it: the expansion phase found a solver seeded from a closed form agreeing with it having never run an iteration. The gadget sweeps march, so the solver does work at every step and the agreement means something.

What is being compared is not the closure residual. The closure residual says the bars are the lengths they should be, and it is at the floor whether or not the gadget is producing the right angle — which is the whole subject of the branch rung. The number quoted here is the departure: the output angle minus the angle the closed form asks for. It is a different quantity and it is the one that can fail.

The ranges are chosen, and the choice is stated #

Each of those sweeps stops short of somewhere, and pretending otherwise would make the measurement worthless.

The reflector has three singular configurations in a turn. When the mirror lies along the input the rhombus flattens onto a line and the two placements of its far vertex merge. When the mirror is perpendicular to the input the far vertex arrives at the pivot and the rhombus closes up on itself; there are two such angles. With the bench’s mirror at $0.8$ radians those sit at $0.8$ and at $0.8 \pm \pi/2$, and the sweep runs between the nearest pair.

The translator’s is one condition: the direction being carried lies along the line between the two points it is being carried between, and the parallelogram has no area. The bench puts its target straight up the page so that angle is a quarter turn away, outside the sweep.

Those are not evasions and they are not hidden. A sweep that crossed a singularity would be measuring the branch change rather than the gadget, and the branch change gets its own figure two rungs on, where the reflector is driven straight through and what happens is recorded. Two measurements, one on each side of a line, is a more informative arrangement than one measurement that averages over it.

Two inputs at one pivot, which is the hidden requirement #

There is a condition on the mean that is easy to miss and governs the whole architecture: its two inputs must be links from the same point.

A rhombus relates the directions of two links that share a vertex. It cannot relate a link at one place to a link at another, because there is no rhombus with a vertex in two places. So every angle a compiled machine wants to combine has to be brought to a common pivot first, and that is a job for the translator rather than for the arithmetic.

The architecture that follows is the one Kempe chose and it is the only one available: do all the arithmetic at one fixed pivot, where every angle is a link from a single point and any two of them can meet in a rhombus. Copying between fixed pivots is free — their separation is a constant, so the parallelogram is always available — so the arithmetic could be spread over several ground points if that were convenient. Copying to anything that moves is not free, and the whole of the summing chain is downstream of that.

It also explains a piece of the compiler’s shape that would otherwise look arbitrary: the arm’s second angle is dragged back to the pivot with a parallelogram before any arithmetic starts. The alternative — doing the arithmetic at the arm’s elbow, which is where $\beta$ naturally lives — fails immediately, because the elbow moves relative to everything except the pivot and its own tip.

Why a rhombus and not something cleverer #

It is fair to ask whether there is a smaller angle-adder, and the answer illuminates why this particular construction is the one built.

There are certainly other mechanisms with angle relations. An antiparallelogram — four bars with equal opposite sides, crossed — has a beautiful angle relation between its input and output, and it is not the one needed: sweeping one gives $\tan(\theta/2)\tan(\psi/2) = (a-b)/(a+b)$, a constant product of half-angle tangents rather than a linear relation. That is the rolling-ellipse relation, and it is exact and useless here, because what the expansion asks for is $m\alpha + n\beta$ and nothing else.

A gear pair multiplies an angle exactly, and gears are outside what this construction is allowed. The whole interest of a universality result for linkages is that it uses only bars and pins — the lower pairs — and a machine that needed gears would be answering a different question.

The rhombus is what is left, and it is enough. Its virtue is that the relation it enforces is linear in the angles, which no other four-bar’s is, and linearity in the angles is exactly what a trigonometric expansion with integer frequencies requires.

Determinate, and what that costs #

All four gadgets have the same bookkeeping property: each adds exactly as many equations as it adds unknowns. A mean is two coordinates and two bars. A reflector is four coordinates, three bars and a rail. An offset is two coordinates and one rigid attachment, which is two equations. A translator is two coordinates and two bars.

So a machine made of gadgets has whatever mobility its arm had, less whatever the closing constraint takes. The arm has two freedoms, the equation takes one, and the machine has one — however many gadgets are hung off it. That is not obvious in advance and it is the reason the construction scales at all: a linkage assembled by adding determinate pieces stays a mechanism rather than becoming a structure.

It is also the reason Grübler’s count is exactly right about an unbraced compiled machine, which is a mild surprise given how often it is wrong about the interesting ones. Every gadget is a dyad or a rigid attachment, none of them is overconstrained, and the count has nothing to miss. That changes, deliberately, when the braces arrive.

Where the arithmetic stops being the problem #

Six bars add two angles. Three bars and a rail double one. A triangular link supplies a phase and a length. Multiplying by $n$ costs about $\log_2 n$ reflectors.

Add all of that up for a curve of degree five and the arithmetic comes to a few dozen bars. The compiled machine for a general quintic has four hundred and thirteen.

The difference is not arithmetic. It is that every angle produced at the pivot has to be carried to where the summing chain needs it, and carrying is the next rung — one parallelogram per hop, one hop per position along the chain, and a chain as long as the number of terms.

The place these gadgets came from #

A note on attribution, because the construction here is not the historical one and it would be dishonest to let the resemblance stand unremarked.

Kempe’s 1876 paper describes an additor, a reversor, a multiplicator and a translator, and gives constructions for each built from contra-parallelograms and parallelograms. The four operations are his and the architecture — compute the angles at a fixed pivot, sum the terms with a chain, close the chain on a line — is his.

The gadgets built here are not. They are a rhombus used two ways, chosen because they are the simplest pieces this site’s solver can position and this site’s gates can check, and because the relation each enforces can be stated in one line and compared against a measurement. Every bar count in this field is a count of these machines. Where the historical figures are mentioned they are described as such, and the comparison is qualitative: Kempe’s own construction for a modest curve runs to numbers of parts that were quoted in the millions and were later brought down a great deal by others, and none of that arithmetic is reproduced here.

What is shared is the result and the shape of the argument. What is measured is this construction, on this site’s solver, and the numbers belong to it.