Crystallography & Symmetry
patterns derived from their groups
Concept

Tetrahedral group — where it appears

The twelve rotations of a regular tetrahedron, which is the alternating group on four letters. It is the smallest of the three polyhedral groups and the only one of them a cubic lattice hosts without its reflections.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The most of an icosahedron a crystal can keep. Every subgroup of the sixty rotations of an icosahedron, found by closure, with the crystallographic ones marked — those whose rotation orders are all among the 1, 2, 3, 4 and 6 that a three-dimensional lattice admits. The largest is 23, of order 12, at index 5; everything containing a fivefold axis is refused. So a crystal containing an icosahedral molecule may fix a twelfth of the molecule's own symmetry and no more, and the remaining 5 orientations have to be related by something other than the site's symmetry.

The most of an icosahedron a crystal can keep

C₆₀ sits in crystals and virus capsids sit in crystals, and neither of them stops being icosahedral. What a lattice can fix is a subgroup — and the largest crystallographic subgroup of the sixty rotations has order twelve, at index five. The five are Kepler's five cubes.

restriction · Local symmetry Which Schläfli symbols close. Every {p, q} with p polygons round each face and q faces round each vertex, from three to six of each. A solid exists only when 2p + 2q − pq is positive, which is the same statement as 1/p + 1/q > ½; the five that qualify carry their vertex, edge and face counts, and the three on the diagonal where the expression vanishes are the three regular tilings of the plane. Past them the expression is negative and the answer is the hyperbolic plane, where the list never ends. The five, the three and the infinity are one inequality read at its three signs.

Five solids from one inequality

Five families of rotation group in space, five regular solids, three regular tilings of the plane and an endless supply of hyperbolic ones — all of it is 1/p + 1/q compared with a half, read at its three signs.

restriction · Finite groups Every four-colouring of the seventeen, by the group that permutes the colours. The 281 subgroups of index four across the seventeen plane groups, each a colouring of a pattern in four colours that the pattern's symmetries permute transitively, sorted by the permutation group they induce on the colours: ℤ₄ 20, Klein 77, D₄ 164, A₄ 8, S₄ 12. The groups without a three-fold rotation use only the cyclic, Klein and dihedral groups; the five with one use only A₄ and S₄, apart from a single Klein colouring of p6m.

Four colours, five ways to permute them

Two colours can be swapped in one way and three can be permuted in two. Four colours can be permuted transitively by five different groups, and the seventeen plane groups use all five: 281 four-colourings sort into 20 cyclic, 77 Klein, 164 square, 8 tetrahedral and 12 fully symmetric. The last twenty belong to the five groups with a three-fold rotation, and they are the lattice reduced modulo two.

classification · Colour

Named alongside it

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

Crystallographic restrictionIcosahedral groupOrbitAbelianisationArchimedean tilingColour symmetryCosetDualityEnumerationThe Euler characteristicFinite groupFullerene

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