Orbifold notation, the shorter language

Three notations name the seventeen wallpaper groups, and each makes a different fact obvious. Hermann–Mauguin makes the directions obvious. Schoenflies makes the abstract group type obvious. Orbifold notation makes the thing that is hardest to see any other way obvious: whether two groups are actually different.

The idea is one move: stop thinking about the pattern and think about the pattern divided by its symmetry. Every point that the group can carry to another becomes one point. What is left is called an orbifold, and its shape determines the group completely.

What the folding produces #

Take a p4 pattern — fourfold rotations, no mirrors — and identify every point with all its images. A generic point has four images, so four points of the plane become one. But the fourfold centres have only themselves: the rotation fixes them, so they do not get glued to anything.

The result is a sphere with three special points on it, of orders 4, 4 and 2. Special how? Walk a small circle round one of them on the folded surface and it closes after a quarter of a turn rather than a full one, because three quarters of the circle were glued away. Those are cone points, and the notation names p4 as 442: a list of the orders of its cone points.

A mirror does something different. Points on a mirror line are fixed by the reflection, so they do not glue in pairs; instead the folded surface acquires a boundary along them, like the edge of a piece of paper. The notation writes * for a boundary, and the numbers after it are the orders of the corners where two mirrors meet.

So p6m — sixfold rotations and every mirror the lattice permits — folds to a triangle with corners of orders 6, 3 and 2, and its symbol is *632. The pattern’s whole classification is a triangle with three angles.

Reading the symbols #

Four kinds of character appear, and each names a feature of the folded surface.

A digit before any * is a cone point of that order: a rotation centre with no mirror through it.

A * starts a boundary. The digits after it are the corners along that boundary, in order, and a corner of order $n$ is where two mirror lines cross at $\pi/n$.

A digit after a * is therefore a rotation centre that does lie on mirrors, which is why it appears as a corner rather than a cone.

An × is a cross-cap: what a glide with no mirror produces, gluing a strip to itself with a reversal. The two groups whose symbols end in × are pg (××) and pgg (22×).

Two examples make the reading concrete. The symbol 4*2 — p4g — says: a fourfold cone point that is not on any mirror, then a boundary with one corner of order 2. And *442 — p4m — says: no free cone points at all, and a boundary with corners of orders 4, 4 and 2. Both groups have eight operations per cell and the same point group, and their symbols are visibly different, which is the notation’s whole claim to usefulness.

The awkward pair, at last obvious #

The clearest case for the notation is p3m1 and p31m — same lattice, same point group, same number of operations, and a distinction that takes several paragraphs to explain in Hermann–Mauguin.

In orbifold notation they are *333 and 3*3.

The first says: no free cone points, and a boundary with three corners of order 3. Every threefold centre lies on mirrors. The second says: one free threefold cone point, then a boundary with one corner of order 3. Some threefold centres lie on mirrors and one kind does not.

Two different surfaces, so two different groups, and no calculation was required beyond reading the symbols. That is what “makes the distinctness obvious” means, and it is the reason this notation is worth learning even by somebody who will keep using Hermann–Mauguin for everything else.

The cost, and why it comes to two #

The notation carries a proof with it, and the proof is arithmetic rather than case analysis.

Assign a cost to each feature of the orbifold:

A symbol describes a wallpaper group exactly when its costs sum to $2$. Check it on *632: the boundary costs $1$, and the corners cost $5/12 + 2/6 + 1/4$, which is $5/12 + 4/12 + 3/12 = 1$. Total $2$. Check it on 442: the cone points cost $3/4 + 3/4 + 1/2 = 2$. Total $2$.

So the classification of the seventeen becomes the problem of listing every way to make $2$ out of those pieces, subject to the orders being whole numbers of at least $2$. That is a small search, it terminates, and it produces exactly seventeen answers.

The sum has a meaning. It is the Euler characteristic of the orbifold, in the form that keeps track of the cone points, and the condition “equals 2” is the condition that the orbifold is flat — that the pattern lives on the ordinary plane rather than on a sphere or a hyperbolic plane. Symbols costing less than 2 describe the finite groups of the sphere; symbols costing more describe the hyperbolic groups, of which there are infinitely many. One arithmetic condition classifies all three geometries, and only the middle case is finite, which explains why seventeen is a number at all rather than a symptom of not having looked hard enough.

The seventeen, in both notations #

Set side by side, the two naming schemes are doing visibly different jobs.

Every row costs exactly two, and checking a few by hand is the fastest way to make the rule stick. The pattern down the table is the notation’s real advantage: the groups pair up as X and *X, with the starred one being the version where the rotation centres have acquired mirrors — and the two exceptions to that pairing, p4g and p31m, are exactly the two awkward cases the other notations struggle with.

The shape of the domain, and what it becomes #

Folding is easier to picture with the domain in front of it, and two cases are worth walking through.

p6m folds to a triangle. Its domain is a twelfth of the cell, bounded on all three sides by mirror lines, and folding does nothing to it except declare those three sides to be boundary. The result is a triangle with corners of orders 6, 3 and 2 — the symbol *632 read directly off the picture. Nothing is glued to anything, because every edge was already a mirror.

p1 folds to a torus. Its domain is the whole cell and none of its edges is a mirror, so the folding glues left edge to right and top to bottom. That is the standard construction of a torus from a square, and the symbol o is the notation’s way of writing “one handle, nothing else”.

Setting that beside the p6m domain at the top of this essay gives the two ends of the notation in one comparison. p6m’s domain is a twelfth of the cell and every one of its three edges is a mirror, so the folding declares all three to be boundary and glues nothing — the result is a triangle with three corners, *632. p1’s domain is the whole cell and none of its edges is anything, so the folding glues all four and declares no boundary — the result is a torus, o. Every other group is a mixture of those two behaviours, edge by edge, and the symbol is the record of which edges went which way.

Between those extremes sit the groups whose domains have some mirror edges and some glued ones, and the symbol records which is which. That is the entire content of the notation: a mirror edge becomes a boundary and a glued edge does not, and everything else follows from counting.

What the site computes, and what it does not #

Honesty about the machinery here matters more than usual, because it would be easy to imply more than is true.

The fundamental domains on this page are computed: sampled on an exact rational grid, one representative taken from each orbit, and the partition checked so that every point’s orbit meets the domain exactly once. That is a real computation and it is what makes a domain figure evidence rather than illustration.

The orbifold symbols on this page are quoted, and they are derived elsewhere. Folding a domain into a surface and reading off its cone points and boundaries is a construction this page does not perform; seventeen dollars does perform it, from each group’s own operations — cone orders from the stabilisers of points on no mirror, corner orders from the stabilisers of points on one, the boundary curves by joining mirror lines that cross, and whatever cost is left over paid in cross-caps — and requires the map from the seventeen groups to the seventeen symbols to be one-to-one, which is the half of the check that catches a mirror miscounted as a glide. What this page supplies on its own is the domain’s area: the cost formula predicts that the domain of a group of order $|G|$ is $1/|G|$ of the cell, and the measured fractions agree with that for all seventeen.

That is a partial corroboration and it is worth naming as partial. A wrong symbol with the right total cost would pass it — and that is not hypothetical: the derivation linked above once read every glide as a mirror, and every one of the seventeen still cost exactly two, because the boundaries the miscount invented were paid for by the cross-caps it stopped seeing. What broke was that three groups came back with another group’s symbol. Cost is the cheap check and injectivity is the one that catches the error, which is why the area agreement here is worth having and is not worth mistaking for a verification of the symbols.

Where the exactness stops #

Two limits, one technical and one about scope.

The notation is complete for two dimensions and awkward beyond it. Conway extended it to the three-dimensional space groups, and the extension exists and is used, but the symbols are longer and the arithmetic condition is no longer a single sum. The two hundred and thirty are still named in Hermann–Mauguin by essentially everybody, and the practical reason is that the crystallographic symbol carries directional information that an experiment measures and the orbifold symbol does not.

The notation says nothing about the lattice. 442 names p4, and p4 lives on a square lattice — but the symbol does not say so, and it does not say how large the cell is or which way it points. That is deliberate: the orbifold is what remains after all of that has been divided out. It is the right notation for asking whether two patterns have the same symmetry type and the wrong one for indexing a diffraction pattern, and confusing the two jobs is the only real hazard in using it.

The surprising part #

The strongest thing about this notation is that it makes the seventeen look like an accident of dimension rather than a fact about wallpaper.

Once the classification is “which symbols cost exactly 2”, the neighbouring questions ask themselves. Symbols costing less than 2 are the finite groups — the rosettes, the point groups, infinitely many because a cone point can have any order. Symbols costing more than 2 are the hyperbolic groups, also infinitely many, and they are the symmetry groups of Escher’s Circle Limit prints. The seventeen sit at a single value, and they are finite in number precisely because they are a boundary case.

The connection worth carrying is to the crystallographic restriction. In the lattice argument, the restriction to orders 1, 2, 3, 4 and 6 comes from an integer trace. In the orbifold argument, no lattice is mentioned at all, and the same five orders fall out of which cone-point costs can be assembled into exactly 2. Two arguments with nothing in common reaching the same five numbers is the sort of agreement that makes a result feel less like a fact about periodic patterns and more like a fact about arithmetic.

The sum is a curvature, and the rule is Gauss–Bonnet #

The cost table looks like a scoring system somebody chose, and it is not. Every entry is a measurement of curvature, and the requirement that the total come to two is a classical theorem with an orbifold’s version of a surface’s area in it.

For an ordinary closed surface, the Gauss–Bonnet theorem says the total curvature is 2π times the Euler characteristic — a fixed number, independent of how the surface is bent. An orbifold has an Euler characteristic too, computed the same way but with each cone point and each corner contributing a fraction rather than a whole, because a cone of order n is a point that only 1/n of a disc’s worth of surface surrounds.

Work that out and the cost of each feature is exactly 2 minus its contribution to the orbifold Euler characteristic. The requirement that the costs sum to two is the requirement that the characteristic be zero, and a characteristic of zero is precisely the condition for a surface to be flat — which is what a quotient of the Euclidean plane has to be.

So the arithmetic is not a coincidence and it is not a device. It is Gauss–Bonnet, applied to an object whose curvature is concentrated at finitely many points, and the three regimes fall out of the sign of one quantity. A total under two leaves positive characteristic: a sphere, and a finite group. A total over two leaves negative characteristic: hyperbolic. And the order of a finite group is recoverable from the same number — a spherical orbifold’s group has order 2/(2 − cost), so a symbol costing 2 − 2/12 names a group of order twelve, with no construction required.

Why the symbol is a complete name #

The claim that the symbol names the group is stronger than the claim that it distinguishes the seventeen, and it is worth separating the two.

A compact two-dimensional orbifold is determined, up to the obvious equivalence, by exactly the data the symbol lists: the genus, how many boundary circles it has, how many cross-caps, and the orders of its cone points and corners in order round each boundary. That is the classification of surfaces with the orbifold data attached, and it says the symbol is a complete invariant — not a label that happens to be different for each of the seventeen, but a description from which the orbifold can be rebuilt.

And an orbifold determines its group, because the group is the orbifold’s fundamental group in the orbifold sense and the plane is its universal cover. So symbol determines orbifold determines group, with no step that could fail for a larger list.

That is what makes the extension upwards free. A notation whose completeness rests on a case-by-case check of seventeen entries would say nothing about the hyperbolic case; one resting on the classification of surfaces says the same thing for every total, which is why the list above two can be enumerated by writing symbols rather than by constructing groups.

The cost of that completeness is the limitation this page already names. The symbol describes the quotient and the quotient has forgotten the lattice — it knows that p4’s orbifold is a sphere with cone points of orders 4, 4 and 2, and it does not know how large the cell is or which way it points, because neither survives the folding.

Who invented it #

The notation is John Conway’s, developed through the 1980s and 1990s and set out fully in The Symmetries of Things, written with Heidi Burgiel and Chaim Goodman-Strauss and published in 2008. Conway also supplied the walking names for the frieze groups — hop, step, sidle, jump — from the same instinct that a notation should be sayable.

The mathematics underneath is older. Orbifolds as objects were introduced by Satake in 1956 under the name V-manifolds, and were developed by Thurston in the 1970s as part of his programme on three-dimensional geometry; the word “orbifold” is Thurston’s, chosen by his students in a vote. The magic-theorem proof of the seventeen is Conway’s own, and it is the shortest complete proof of the classification in print.

Where the ladder goes next #

The notation this one is an alternative to is Hermann–Mauguin, which stays the working standard for good reasons.

The object being folded is the fundamental domain, and the theorem the arithmetic proves is the classification.

The pair it settles fastest is p3m1 and p31m, and the list it names is the seventeen.

What the pictures here cannot show. An orbifold is a surface obtained by gluing, and none of the figures on this page shows one — they show fundamental domains, which are what an orbifold looks like before the gluing. The gluing is where all the content is: whether two edges of the domain are joined directly or with a reversal is the difference between a boundary and a cross-cap, and a drawing of the unglued piece cannot say which.