Seventeen — the series · Crystallography & Symmetry

The seventeen

Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.

part 1 · classification

The wallpaper group p4m. A pattern with the symmetry of p4m, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.

Reading Hermann–Mauguin

p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.

part 2 · classification

The wallpaper group p3m1. A pattern with the symmetry of p3m1, generated by applying the group's 6 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.

Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence.

part 3 · classification

The seventeen sorted by lattice: 2, 5, 2, 3, 5. The five plane lattices, each drawn from the basis every other figure here uses, with the wallpaper groups that sit on it and the order of each against its lattice's holohedry. The counts are 2, 5, 2, 3, 5, which is seventeen again, arrived at by a different route from the case analysis on rotation order. Two relations hold and both are checked. Every group's order divides its lattice's holohedry, because an operation has to map the lattice onto itself before it can map the pattern onto itself — which is why a quarter turn has nowhere to live but a square lattice. And the converse fails on every one of the five: each lattice carries at least one group whose order falls short of what the lattice offers, so knowing the lattice narrows the group to a handful of candidates and never to one. The pairs printed in the accent colour are the groups that take everything their lattice permits.

The classification proof, one branch at a time

Seventeen is a theorem, and the argument that establishes it is a finite case analysis that fits on a few pages. Working through it is the difference between knowing the number and knowing why there is no eighteenth.

part 4 · classification

A fundamental domain for p6m. One representative from every orbit of p6m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

Orbifold notation, the shorter language

Fold a pattern up along its own symmetries and what remains is a small surface with marked points. Its shape is a complete name for the group, and reading the name off costs an arithmetic sum that has to come to two.

part 5 · classification

The seventeen signatures, and the seventeen groups. Every combination of features costing exactly two, beside the plane group each one names. The left column is produced by an accounting identity that has never heard of a lattice; the right by reading seventeen groups' own operations — their rotation centres and orders, which of those lie on mirrors, and how many closed curves the mirror lines make once equivalent lines are identified. The map between the two lists is a bijection, and the figure does not appear unless it is one — in both directions. A signature with no group and a group whose signature is not on the list are both refused, and so is the failure that actually happens: two groups deriving one signature, which costs exactly two and passes every check but injectivity.

Seventeen dollars

Conway's magic theorem prices the features a folded-up pattern can have — a handle costs two, a mirror boundary one, a cone point of order n almost one — and requires the total to come to exactly two. There are seventeen ways to pay, and the classification falls out of an accounting identity that never mentions a lattice.

part 6 · classification

The (2, 3, 7) group, in the Poincaré disk. A triangle with angles π/2, π/3 and π/7, reflected in its own three sides until depth 12: 380 triangles, alternating in handedness because every generator is a reflection. The sum 1/2 + 1/3 + 1/7 is less than one, so the triangle does not fit in the flat plane and the drawing is of the hyperbolic one, with the whole plane squeezed inside a disk. Every triangle has the same hyperbolic area; the ones near the edge look small because the model shrinks distances there, and the tiling stops at the edge of the drawing rather than at the edge of anything.

Past two, the list does not stop

Conway's accounting says a wallpaper group costs exactly two dollars, and there are seventeen ways to spend it. Spend less and the answer is a finite group. Spend more and the list is infinite — but the cheapest thing past two costs two and one eighty-fourth, and nothing at all lies in between.

part 7 · classification

Every group decided by a window of radius 1. For each of the seventeen, the radius at which a round window on the pattern admits exactly the group's own operations and no others — with the numbers it admits at each smaller radius beside it. Two opposite failures are visible. Most groups under-report at small radii, because an operation carrying points out of the window cannot be tested at all; cm over-reports, admitting operations the pattern does not have. The groups that take longest to settle are the ones distinguished by a glide, which moves a point half a cell before anything can be compared.

How much pattern is enough

Every claim here about a pattern's group is a claim about an infinite pattern. A reader sees a patch. Measuring what a finite window can decide gives a number — about one cell's radius — and two opposite ways of being wrong on the way there.

part 8 · classification

p4m from 3 orbits of waves — detected p4m. A density built as a sum of 3 symmetry-adapted waves of p4m, each of them the average of a plane wave over the group, shaded from light to dark across one cell. The level set of this density — the darkest points of it — was handed to the same detector the pattern figures use, and it reports p4m, which is exactly the group the waves were built from. The waves are invariant by construction, so the density can never have less symmetry than the group; the interesting direction is the other one.

How many waves a group permits

A pattern can be written as a sum of waves instead of as an orbit of a motif, and then the group ties the coefficients together and forbids some outright. Building a density from the permitted ones and handing it back to the detector closes the same loop through a different door — and at low resolution the density has symmetry the crystal has not.

part 9 · classification