Crystallography & Symmetry
patterns derived from their groups
Concept

Icosahedral symmetry — where it appears

The symmetry of an icosahedron: sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden and the one the first quasicrystal turned out to have.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Every solution of the axis equation. The integer solutions of 2 − 2/N = Σ(1 − 1/nᵢ), which is what counting the pairs (rotation, fixed pole) two ways gives. Two classes of axis force n₁ = n₂ = N and give the cyclic groups; three classes give the dihedral family and exactly three sporadic answers — (2, 3, 3), (2, 3, 4) and (2, 3, 5), of orders 12, 24 and 60, which are the rotation groups of the tetrahedron, the octahedron and the icosahedron. Four classes are impossible, because four terms of at least a half already exceed the left-hand side. Nothing about crystals has been used.

Before the lattice has a say

Every finite group of motions of the plane is a Cₙ or a Dₙ, and every finite group of rotations of space is one of five families. Both lists come out of counting rather than out of crystallography — and then the crystallographic restriction deletes almost all of them, leaving eleven.

restriction · Finite groups A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call.

What Shechtman measured

A diffraction pattern with sharp spots and tenfold symmetry, in April 1982. Sharpness meant order and tenfold meant no lattice, and the two had been believed inseparable — so the interesting question is what the observation had to rule out before it could mean anything.

aperiodic · Quasicrystals The icosahedral group, counted. The sixty rotations of an icosahedron, found by trying every map that sends one adjacent pair of vertices to another and keeping those that carry the whole vertex set onto itself. They fall into 6 axes of order 5, 10 axes of order 3, 15 axes of order 2 — and the fivefold axes are the reason this group cannot be the point group of any crystal, since no three-dimensional lattice admits a rotation of order five. Quasicrystals have it anyway, which is what made 1982 an argument rather than a measurement.

Icosahedral symmetry

Sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden, and the one the first quasicrystal turned out to have.

aperiodic · Quasicrystals 60 vertices, 12 pentagons. A closed net with three edges at every vertex: 60 vertices, 90 edges and 32 faces, of which 12 are pentagons and 20 are hexagons. The pentagons are picked out in the second colour. Their number is not a property of this cage — it is twelve for every closed trivalent net of pentagons and hexagons, at any size, and the hexagon count is free.

Twelve pentagons, and no way round them

The crystallographic restriction forbids a five-fold face in a flat repeating net. Curve the net into a closed cage and the same three lines of arithmetic require exactly twelve of them — at any size, with the hexagon count free. What a lattice forbids, closing up compels.

restriction · Curvature 5 units of 70.53°: 7.36° left. 5 tetrahedral units of face-centred cubic metal, each the mirror image of its neighbour in a {111} plane, arranged about a common ⟨110⟩ edge. The angle between two such planes is arccos(1/3) = 70.53°, computed from the plane normals rather than quoted, and 5 of them come to 352.64°. The shaded sector is what is left over: 7.36°, or 2.04 per cent of a full turn, which must be taken up by strain, by a gap, or by a defect along the axis.

Five copies, and the gap they leave

Gold, silver and silicon grow particles with a five-fold axis down the middle, out of a lattice that forbids one. Nothing is violated: five tetrahedral pieces of ordinary face-centred metal, each the mirror image of its neighbour, come to three hundred and fifty-two and a half degrees rather than three hundred and sixty — and the seven degrees left over have to go somewhere.

restriction · Local symmetry The fewest contacts twelve pentagons can have, by size. For every cage of pentagons and hexagons up to forty-four atoms, the number of pairs of pentagons sharing a bond. The lower line is the fewest any cage of that size achieves — 30, 24, 21, 18, 17, 15, 14, 12, 11, 10, 9, 8 — the upper line the most, and the dashed line the bound that counting edges gives: the twelve pentagons carry sixty edges between them, a contact uses two and an edge to a hexagon uses one, so the contacts cannot fall below 30 − 3h with h hexagons. The bound is attained while the hexagons are few and goes loose at five, after which each extra hexagon removes about one contact rather than three. The number of cages at each size is printed beneath, and it is the least rather than the average that the bound is about.

How close the twelve must be

The charge fixes twelve pentagons and says nothing about where they go, because it is a sum over faces and cannot see which face touches which. What it cannot see is a graph on twelve points, and the fewest edges that graph can have falls from thirty to eight over the cages a census reaches — then keeps falling at a rate that puts its first zero exactly where the truncated icosahedron is.

restriction · Curvature

Named alongside it

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

Crystallographic restrictionCountingFullereneQuasicrystalTrivalent netCensusClosed surfaceClosureCombinatorial curvatureCoordination numberCurvatureCyclic group

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