Crystallography & Symmetry
patterns derived from their groups
Concept

Free action — where it appears

A group action in which no operation but the identity leaves any point where it is. Folding a pattern along a group acting freely gives a surface with no marked points, which happens for exactly two of the seventeen plane groups and thirteen of the two hundred and thirty space groups.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided.

Ten ways for space to be flat

Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

space-groups · Flat space Two vertices and three edges: two nets, at every box size tried. Every net with two quotient vertices and the stated number of edges, counted inside boxes of voltages of several sizes. One cross voltage is set to zero by the gauge — the freedom that moving one vertex into another cell gives — and the rest are drawn from the box. Each entry is the count of nets whose placement separates their vertices, plus the count of those whose does not: the first has a canonical description and stops growing, and the second does not have one and therefore keeps rising with the box. The reducible column is the descriptions thrown away for a reason the one-vertex census never had — cycles generating the whole of ℤ² and a net whose own cell holds one vertex rather than two — and it is empty at every odd edge count, because the swap that would reduce a description pairs its edges and an odd number cannot pair.

Every net with two vertices, counted

The one-vertex census could not contain the honeycomb, because the honeycomb has two vertices in its cell. Adding the second one closes a family at two nets, removes the floor of p2 entirely, makes a third of the members undrawable, and forces the census to refuse a kind of description the first one never met: an honest quotient graph written on twice the cell it needs.

applied · Nets The sphere fixes a count; the torus fixes only a difference. Euler's relation for a trivalent net gives Σ (6 − n) pₙ = 6χ, so the surface fixes one linear combination of the face counts and nothing else. On a sphere that combination is twelve, which with no face smaller than a pentagon forces exactly twelve pentagons. On a torus it is zero, which permits any number of pentagons provided as many heptagons pay for them — and permits none at all, which is the plain hexagonal net. On a surface of two holes it is minus twelve, so heptagons become compulsory instead.

As many heptagons as pentagons

A trivalent net on a sphere must have exactly twelve pentagons. The same three lines of arithmetic on a torus give zero — which does not forbid pentagons, it makes them pay: every pentagon has to be balanced by a heptagon, and the counts are otherwise free. One rotated bond in a wrapped honeycomb makes two of each and changes nothing else.

restriction · Curvature Every closed surface, and the two that charge nothing. The same accounting indexed by Euler characteristic rather than by genus. An orientable surface has χ = 2 − 2g, so it only ever occupies an even row; a non-orientable one has χ = 2 − k and occupies every row from one downwards. The odd rows therefore belong to surfaces that cannot be oriented and to nothing else — and the first of them, the projective plane, charges six. Six pentagons is a bill no orientable surface presents.

The surfaces a count by genus skips

A count indexed by genus steps in twelves and lands only on even numbers. A closed surface can have any characteristic at or below two, and the odd ones belong to the surfaces that cannot be oriented — where the projective plane charges six pentagons, a bill no orientable surface ever presents.

restriction · Curvature Where the sphere and the projective plane have no net. The number of different closed nets with three bonds at every atom and faces that are pentagons and hexagons only. On the sphere, with twelve pentagons and k hexagons for k up to 12, every count has at least one net except k = 1. On the projective plane, with six pentagons and h hexagons, each count sits under the sphere count it lifts to, since every hexagon of a projective net becomes two on the sphere. The projective counts for h = 0 to 6 are 1, 0, 0, 1, 1, 3, 3, so the projective plane has no net at h = 1 or 2: two gaps where the sphere has one. Every sphere count was found by enumeration and agrees with the published one.

A gap the sphere does not have

A net of pentagons and hexagons on the projective plane must have six pentagons, and the count permits any number of hexagons. Not every number happens. Lifting each net to the sphere turns the question into one about which cages have a centre — and the answer leaves two gaps where the sphere has one.

restriction · Curvature A centre at every other ring, and never between. Two families of closed cage, each a tube of hexagons closed at both ends by a cap of six pentagons, taken from no rings of hexagons to 8. The top row has five faces to a ring and a pentagon at each pole; the bottom row has six and a hexagon. Each box holds the cage's number of atoms with its number of hexagons beneath, and a box is drawn solid with a dot under it when the cage has a symmetry that reverses orientation and fixes nothing — a centre, which is what lets the cage halve onto the projective plane. The five-family has one at even numbers of rings and the six-family at odd ones, so their hexagon counts are 0, 10, 20, 30 … and 8, 20, 32, 44 … — two arithmetic progressions rather than two rows.

A centre at every other ring

A census cannot settle an infinite row, and the construction proposed to settle it was a tube capped at both ends, lengthened a ring at a time. Carried out, it alternates: a centre appears at every other ring and never between, the two families it permits reach two arithmetic progressions rather than a row, and the first of them opens with exactly the cage the census found could not halve.

restriction · Curvature 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.

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.

operations · Presentations 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.

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.

operations · Presentations

Named alongside it

The objects these essays reach for when they reach for this one.

Crystal netOrientabilityCombinatorial curvatureEnumerationInversion centreTorsionAutomorphismCayley graphThe Euler characteristicFixed pointFullereneGraph isomorphism

All concepts