Crystallography & Symmetry
patterns derived from their groups
Concept

Automorphism — where it appears

A symmetry of a structure regarded as a map of the structure to itself, preserving whatever relation defines it. For a graph it permutes vertices and carries edges to edges; whether it also becomes a motion of the plane is a separate question, answered by the equilibrium placement.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Rotation orders 1, 2, 3, 4, 6 and no others. Every net in this collection, with the orders of the rotations its own symmetry group has, and the degrees of its vertices beside them. The orders are 1, 2, 3, 4, 6 — the crystallographic restriction, arrived at with no length anywhere in the argument: the translations of a net are ℤ² by construction, an automorphism carries translations to translations, so it acts on ℤ² by an integer matrix, and an integer trace in the interval from minus two to two is one of five numbers. The degree column is there because the two are constantly confused: a net may perfectly well have vertices of degree five, and one here does.

The restriction, with no lattice assumed

The proof that only two-, three-, four- and six-fold rotations are possible is usually stated about a lattice, and every step of it turns out to need no lengths at all. A periodic graph has the same theorem, proved the same way — and a graph may have a five-fold symmetry the plane cannot receive.

restriction · Restriction the honeycomb net: cmm against p6m. the honeycomb net drawn twice. On the left a placement chosen by hand, whose symmetry group is cmm of order 4; on the right the placement in which every vertex sits at the average of its neighbours, whose group is p6m of order 12. The graph is identical in the two — the same vertices joined the same way — so every symmetry of the left-hand drawing is a symmetry of the net and the right-hand drawing has them all. Each detected operation is then required to carry every edge of the quotient graph to an edge, which is what makes it a symmetry of the net rather than of the point set.

The placement nobody chose

A net has no coordinates, so drawing one means inventing them. There is exactly one way to invent them that involves no choice: put every vertex at the average of its neighbours. The drawing that results has the largest symmetry group the net admits, and this site's own detector finds it.

applied · Nets 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

Named alongside it

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

Combinatorial curvatureCrystal netEnumerationFree actionFullereneInversion centreOrientabilityAccidental symmetryBarycentric placementClosed surfaceCrystallographic restrictionDegree

All concepts