Crystallography & Symmetry
patterns derived from their groups
Concept

Mirror — where it appears

A reflection thought of as a line or a plane rather than as an operation: the fixed set of the reflection, which every point is carried across. A pattern's mirrors are the places its picture folds onto itself, and a group's are what a fundamental domain is bounded by.

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

The five kinds of glide plane. All five glide letters: a, sliding by a/2; b, sliding by b/2; c, sliding by c/2; n, sliding by (a+b)/2, (b+c)/2 or (a+c)/2; d, sliding by (a+b)/4, (b+c)/4 or (a+c)/4. 3 of those drawn are axial — the slide is half of a single cell vector lying in the plane. A glide's square is a pure translation, so its slide can only be half of a lattice vector, and the quarter-cell d glide exists only where centring has already made the half-diagonal a lattice vector.

Reflect, then slide by half of something

A glide's slide must double to a lattice vector, which leaves three candidates in any plane and a fourth that exists only where centring has already made a half-diagonal into a lattice vector. Five letters, and the fifth is the one the enumeration explains.

space-groups · Screws and glides 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

Named alongside it

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

Cartan integerCentringCrystallographic restrictionDiamond glideDihedral groupEnumerationGlide planeIntrinsic translationLatticeLattice translationPoint groupReflection

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