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”.

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 themselves are not computed here. Folding a domain into a surface and reading off its cone points and boundaries is a construction this site’s machinery does not perform, and the symbols quoted above come from the literature. What the site does supply, and what makes the quotations checkable rather than decorative, 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. Saying so is cheaper than the alternative, which is a reader assuming the symbols were derived here and treating them as verified.

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.

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.