Crystallography & Symmetry
patterns derived from their groups
Generator

Seventeen groups, leading coefficients 2, 4, 8, 9, 16, 18, 36

Seventeen groups, leading coefficients 2, 4, 8, 9, 16, 18, 36
Seventeen groups, leading coefficients 2, 4, 8, 9, 16, 18, 36. Every one of the seventeen measured against the same kind of generating set: the two lattice translations and the point-group generators. The ball sizes are fitted to a quadratic on residue classes and accepted only when the fit is exact, so a period of one means the counts are a plain polynomial from the tail onwards. The leading coefficient turns out to be the order of the point group times a number that depends only on the shape of the lattice — two where the generators make a square ball and three where they make a hexagonal one — which is as close as growth comes to seeing geometry. It is not an invariant of the group: change the generating set and it changes.

Every one of the seventeen measured against the same kind of generating set: the two lattice translations and the point-group generators. The ball sizes are fitted to a quadratic on residue classes and accepted only when the fit is exact, so a period of one means the counts are a plain polynomial from the tail onwards. The leading coefficient turns out to be the order of the point group times a number that depends only on the shape of the lattice — two where the generators make a square ball and three where they make a hexagonal one — which is as close as growth comes to seeing geometry. It is not an invariant of the group: change the generating set and it changes.

5 essays call cayley. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 522 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

p1, p2, p4, p6m: every one quadratic. How many elements each group has at word length at most R, to 14 terms, against the same kind of generating set. Every curve is a quadratic in R — which is the group knowing its own dimension, since a crystallographic group of d dimensions grows like R to the d and nothing about the counting mentions the plane. The curves differ by a factor: p1 reaches 421, p2 reaches 786, p4 reaches 1464, p6m reaches 5478. Operations

How fast a group grows

Take a wallpaper group, forget the plane, and keep only the generators and the rule for multiplying. Count the elements that can be spelled in at most R letters. The answer grows like R squared — for every one of the seventeen — and the group has told you the dimension of a plane it no longer knows about.

 p4: a domain of 38 cells with 7 walls. The fundamental domain of p4 on a grid of 12ths, with the walls it shares with its neighbouring copies marked. Each wall names the element that carries this copy onto the copy across it, and there are 7 distinct such elements. Those elements generate the whole group — checked by closing them up and requiring every coset and the whole translation lattice to be reached, not assumed — which is Poincaré's theorem, and it means the generators of a wallpaper group can be read off a picture. The domain is pixelated rather than a polygon, so the wall count is a property of this domain and not of the group. Operations

Every wall names a generator

The copies of a fundamental domain tile the plane and stand in one-to-one correspondence with the elements of the group. So the elements that carry the home copy across a wall generate everything — and the generators of a wallpaper group can be read off a picture rather than looked up.

 One curve falls and the other does not. The boundary's share of a ball, against the radius, for a plane group and for the free group on two generators. The plane group's falls like one over the radius and goes to zero; the free group's rises to two thirds and stays. A group with no sequence of regions whose boundary becomes negligible has no shape-independent average, and that is not a difficulty in the analysis — it is a property of the group. Operations

The boundary a growing region forgets

Quoting a density assumes the region it was averaged over does not matter, and that assumption is a property of the group of translations rather than of the crystal. A ball in a plane group grows like R² and its boundary like R, so the edge becomes negligible — and where that fails, the average genuinely moves. The free group on two generators keeps two thirds of itself on the boundary forever, and a slab seven layers deep is wrong by exactly one seventh however wide it is made.

 The square net drawn as the Cayley graph of four different groups. The same square net four times. In each panel the edges are coloured by the group operation that spells them, read from one regular subgroup of the net's symmetry group: the lattice of translations, a p2 whose half-turn centres sit at the midpoints of edges, a pmm whose four mirrors run between rows of vertices, and a p4 whose quarter-turn centres sit in the middle of the squares. No symmetry element drawn passes through a vertex, which is what makes each group act without fixing any, and the large dot is the vertex named by the identity. Operations

The square net draws eleven groups

The Cayley graph of a wallpaper group is a periodic net. Asked the other way, the question is which nets are Cayley graphs, and of what. A net is one exactly when some group of its symmetries carries every vertex to every other and fixes none. All eleven uniform nets pass. The square net is the Cayley graph of eleven of the seventeen groups, the kagome net of exactly one, and between them the eleven nets use all seventeen.

 Every special position of the seventeen plane groups, with its regular subgroups. The 55 special Wyckoff positions of the seventeen plane groups, one chip each, labelled by site symmetry and points to a cell and carrying the number of subgroups that act on the position regularly. None is zero. 26 positions have one point to a cell and are reached by the lattice alone; 12 can be reached only by subgroups containing rotations; four, the positions of three points to a cell in p3m1, p31m, p6 and p6m, are reached by exactly one subgroup each. Operations

No Petersen graph in the plane

A finite graph can look the same from every vertex and still be the picture of no group; the Petersen graph is the smallest that does. A periodic net in the plane cannot. Whether one is a Cayley graph depends only on where its vertices sit, and they sit on one of the fifty-five special positions of the seventeen plane groups or on a general one. Every one of the fifty-five has a subgroup that carries any of its points to any other in exactly one way, and four of them have exactly one.

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