Crystallography & Symmetry
patterns derived from their groups
Concept

Root system — where it appears

A root system is a finite set of vectors closed under the reflections in its own members, whose lengths and angles obey strict integrality conditions. They are classified completely, and the lattices their vectors generate include the most symmetric lattices in each dimension.

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

Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate.

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

restriction · Restriction Every form tried, by its symmetries and by how well it packs. 5902 totally positive forms of the ring of fifteenth roots of unity, sorted by the order of their symmetry group into five rows. Forms with 60 and 120 symmetries form scattered bands. Two lattices have 1,440, the tensor products of A2 with A4 and with its dual. One has 14,400, a member of the icosian line; one has 28,800, the icosian ring with its plain trace; and one is E8. The highest Hermite number among forms with only 60 symmetries is above that of the 28,800 lattice.

One ring is E8 and a lattice with no mirror

An operation of order thirty as one block in eight dimensions preserves a whole family of forms, and E₈ was one point of it. Walking the family finds sixty symmetries almost everywhere and two lines above that. One is tensor products. The other is the icosian ring, on which E₈ is famously built, under a sliding choice of form: 14,400 symmetries all along it, 28,800 and not one reflection at one end, and at the other end E₈, where the group jumps by a factor of nearly fifty thousand.

restriction · Restriction

Named alongside it

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

Crystallographic restrictionCartan integerCyclotomic polynomialsDihedral groupLatticeLattice automorphismMinimal vectorsMirrorPoint groupReflectionWeyl group

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