The relations a polygon dictates

Every wall names a generator computes one half of Poincaré’s polygon theorem. It colours pixels to build a fundamental domain, walks out of it in the four grid directions, records which operation carries each neighbour home, and checks that those elements generate the group. That is the generators.

The other half is the relations, and that essay says plainly that it does not compute them:

Poincaré’s theorem in its full form also produces the relations — from the cycles of copies round a vertex of the tiling — and that half is not computed here. A pixelated domain’s vertices are pixel corners, so the cycles it produces are artefacts of the grid.

The obstacle is real and it is not about effort. A relation comes from walking round a vertex of the domain, and a domain made of squares has vertices wherever two squares meet — hundreds of them, all of them corners of the grid rather than corners of the group’s own tiling. Any cycle taken round one of those would be a fact about the grid.

So the domain has to be a polygon. This essay builds one.

The Dirichlet domain, which needs no search #

Pick a point p whose stabiliser is trivial — no operation of the group fixes it. Its orbit is a set of points, one per group element, and the Dirichlet domain of p is the set of points at least as close to p as to any other point of the orbit.

That set is an intersection of half-planes, one per orbit point: the side of the perpendicular bisector on which p lies. An intersection of half-planes is a convex polygon and it is computed by clipping rather than by searching, which is the first thing the exact construction buys.

It is also a fundamental domain, for a reason worth stating because it is why the trivial stabiliser matters. Every point of the plane is at least as close to some orbit point as to any other, so every orbit meets the domain; and if two interior points were in one orbit, the element carrying one to the other would carry the domain onto a domain sharing interior with it, which two Dirichlet cells of distinct points do not. A point on a mirror or at a rotation centre breaks this, because its orbit is smaller than the group and its cell is correspondingly larger than a fundamental domain. The construction here refuses such a point rather than quietly producing the wrong polygon.

The area check is worth doing and is done: the domain’s area, divided by the cell’s, must be exactly one over the number of cosets. It comes out right on all seventeen to fifteen decimal places, which is the arithmetic saying the polygon is the right polygon.

Every edge is already an element #

Here is what the exact construction gives for free, and it is the thing that makes the rest short.

An edge of the domain lies on the bisector of p and one particular orbit point g·p. On the far side of that bisector, points are closer to g·p than to p — so the region on the far side is the Dirichlet domain of g·p, which is g applied to this one. The edge is shared between the home domain and g’s copy of it, so the edge belongs to g.

No search, no walking out and reducing back, no ambiguity about which element to record. The pixelated construction has to do all of that; the polygon hands it over.

The pairing follows immediately and is likewise not a choice. The edge e_g lies in the boundary of the home domain D and of gD. Apply g⁻¹ to it: the image lies in the boundary of g⁻¹D and of D, and the edge of D shared with g⁻¹D is e_{g⁻¹}. So the pairing map for e_g is g⁻¹, and it carries e_g to e_{g⁻¹}. The construction checks the geometry anyway — the image of the edge must land on its partner, to within rounding — because a pairing derived correctly and implemented wrongly looks exactly like a pairing derived wrongly.

An edge whose element is its own inverse is paired with itself, and there are two ways for that to happen. A reflection fixes the edge pointwise, and the quotient has a mirror boundary there. A half turn folds the edge at its midpoint, and the quotient has a cone point of order two at that midpoint — which is not a vertex of the polygon at all, and is the thing a walk round the corners would miss.

Walking round a corner #

Now the relations.

A corner is a vertex of the polygon together with one of the two edges meeting there. Apply that edge’s pairing: the corner lands on another corner of the same polygon, because the pairing carries the polygon to itself as a set of edges. At the new vertex take the other edge and apply its pairing. Repeat.

The walk closes — there are finitely many corners and every step is reversible — and the composition of the pairings along it is an isometry fixing the starting vertex. An isometry of the plane fixing a point is a rotation or a reflection; here it is a rotation, of some finite order n which the construction finds by multiplying it out rather than by reading the vertex’s stabiliser.

The relation is that the composition, raised to the power n, is the identity. Poincaré’s theorem is that the edge pairings together with these relations present the group.

Each vertex of the quotient is walked twice, once each way round, and the second walk gives the inverse of the first relation. Keeping both is harmless and is also a trap: a presentation carrying every relation twice is insensitive to one of them being dropped, so the check that matters below would pass on a presentation that was missing something. One relation per vertex of the quotient, and the two walks are gathered by the vertices they visit.

Seventeen presentations #

The result is a presentation for each group: one generator per pair of edges, one relation per vertex of the quotient, and one further relation for each edge the group folds in half — the generator’s square.

The p1 row is worth a sentence, because it is the case everyone knows the answer to. The plane group of pure translations is ℤ², generated by two translations that commute — two generators, one relation. The polygon gives three generators and two relations, and it is not wrong: the Dirichlet domain of a point in a generic lattice is a hexagon, so there are three pairs of opposite edges and three translations that generate, with two vertex relations saying the three compose to nothing round each of the hexagon’s two vertex classes. It is a different presentation of the same group, and the geometry chose it rather than the convention.

The check that actually decides #

Every relation the polygon produces holds in the group — the construction evaluates each one in the concrete isometries and requires the identity. That check is necessary and it is nowhere near sufficient, and it is worth being blunt about why: the relations of a larger presentation hold too. A presentation missing a relation presents a bigger group, and every relation it does carry is still true.

What settles it is coset enumeration. Take the subgroup generated by the translations two cells along each axis. Its index in the plane group is fixed by the geometry: four cells times the order of the point group, and nothing about the presentation enters that number. Then run Todd–Coxeter on the derived presentation over that subgroup — which needs the two translations as words in the derived generators, found by breadth-first search through the Cayley graph — and compare.

The test is only a test if it can fail, so it is run on presentations that should fail: for each group, each relation is dropped in turn and the enumeration re-run. Every one of them runs away — the enumeration exceeds its cap without closing, which is the larger group announcing itself. That is the difference between the necessary check and the sufficient one, and it is visible rather than argued.

What the polygon knew about the orbifold #

There is a second agreement in the results and it was not designed for.

Each vertex cycle has an order, and each folded edge contributes a cone of order two at its midpoint. Collect those numbers and they are the cone and corner orders of the group’s orbifold — the object orbifold notation names and seventeen dollars derives from the group’s stabilisers.

The agreement is not a coincidence and it is not circular. It is the same object described twice: the quotient of the plane by the group is a surface with marked points, and a fundamental polygon glued along its pairings is that surface, so the polygon’s vertices become the surface’s marked points and the orders match. Having both computations makes that statement checkable rather than merely true.

The folded edges are the half of it worth remembering. A cone point of order two can sit at the midpoint of an edge rather than at a corner, and a construction that only walked the corners would produce p2’s orbifold with no cone points at all instead of four. The polygon’s corners are not the whole of its special points, and the case that shows it is the second-simplest group there is.

A different polygon, the same orbifold #

The construction takes a base point, and a base point is a choice. So the polygon is a choice, and so is the presentation — which raises the obvious question of what, if anything, is not.

Moving the base point changes the domain visibly. p4 built round one point is a square with two pairs of edges, giving two generators and three relations; built round another it is a pentagon with three edge pairs, giving three generators and four. p3 goes from a four-sided domain to a six-sided one. p6 from three edges to five. Seven of the seventeen change shape among the base points tried here, and the presentations change with them.

What does not change is the multiset of cone and corner orders. On all seventeen groups and every admissible base point, it comes out the same — which is the statement that those numbers belong to the group and not to the picture.

p4 is the case worth following, because the order-two cone point moves. In the square domain it sits at a corner, and the walk round that corner composes to a half turn. In the pentagon it sits at the midpoint of a folded edge, where no walk goes at all, and it is found by noticing that one edge is carried to itself by a half turn rather than by a reflection. Same group, same orbifold, same number 2 in the answer — arrived at by two mechanisms that look nothing alike.

That is the argument for computing the folded edges rather than only the corners, and it is stronger than the p2 case that motivated it. In p2 all four cone points sit at edge midpoints for every base point, so a construction that only walked corners would report p2’s orbifold as having none and would be visibly wrong. In p4 such a construction would be right for one base point and wrong for another, which is worse: it would look correct until somebody changed a number that should not have mattered.

The invariance is also what makes the whole exercise honest. A construction that produced a different orbifold for a different base point would be measuring its own conventions, and the check that it does not is the same kind of check as a cell being a choice — the description has freedom in it, the object does not, and the way to tell them apart is to move the description and see what stays still.

The relations are the loops in the Cayley graph #

There is a second reading of the vertex cycles that costs nothing and explains where they come from.

Every wall names a generator ends by observing that the copies of the domain, joined when they share a wall, form the Cayley graph of the group with respect to the wall elements — one vertex per copy, one vertex per group element, one edge per generator applied. That is where the generators live.

A relation is a word in the generators equal to the identity, which in the Cayley graph is a closed loop starting and ending at the home copy. There are infinitely many such loops, and a presentation is a finite set of them from which every other can be built by conjugation and composition.

The vertex cycles are exactly the smallest ones. Walking round a vertex of the tiling visits the copies that meet at that vertex, in order, and returns — a loop in the Cayley graph, and a short one, because the number of copies round a vertex is the order of its rotation. Poincaré’s theorem is the statement that those loops generate all the others, which is why the corners of one polygon are enough and why an infinite graph can be described by a finite list.

That also says what the pixelated construction was really missing. It has the graph — it walks between copies perfectly well — but its loops are loops round pixel corners, and those are loops in a subdivision of the graph rather than in the graph. They are true relations, in the sense that they hold; they are simply not the ones that generate.

What this does not do #

It is Poincaré’s theorem used, not proved. That the pairings and the cycle relations present the group is the theorem; what is computed here is the pairings, the cycles, and a verification that the resulting presentation is correct for each of these seventeen groups. Verifying seventeen cases is not a proof and the two are not confused.

It is the plane. The same construction works in three dimensions with polyhedra, where the cycles run round edges rather than round vertices and the bookkeeping is longer; the space groups’ presentations are not computed here. It also works in the hyperbolic plane, and there it is the tool of choice — past two the list does not stop enumerates the triangle groups whose presentations this construction would produce, and the reason it matters more there is that there is no finite list to look them up in.

And the presentations are not the shortest. A generic base point gives a hexagonal domain where a special one would give a rectangle, and three generators where two suffice. Shortening a presentation is a separate piece of work — how few operations make a pattern is that question asked directly — and the polygon’s answer is the one its geometry dictates rather than the one a person would choose.

Why it is worth having #

The generators half of Poincaré’s theorem answers “what does this group do”. The relations half answers when a search of them is finished, and that is the question a presentation exists to answer: two words in the generators are the same operation exactly when one can be turned into the other using the relations. Without them a list of generators is a list of moves with no rule for when two sequences agree, which is telling two words apart with nothing to tell them apart by.

For a crystallographer the practical version is smaller and still useful. A presentation is what lets a computer decide whether two symmetry operations, written as products of generators, are the same one — and the geometry of a domain, which a crystallographer already draws, contains it. The corners of the picture are the relations, and until the picture has corners they are not there to read.