Crystallography & Symmetry
patterns derived from their groups
Concept

Companion matrix — where it appears

The integer matrix that acts as multiplication by x on polynomials modulo a given one, so it satisfies that polynomial and no smaller one. Built from a cyclotomic polynomial, it is the smallest integer matrix that turns a plane by a primitive fraction of a turn.

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

An integer matrix of order 5. The companion matrix of the 5th cyclotomic polynomial has whole-number entries and order exactly 5, so it is a genuine 5-fold symmetry of a 4-dimensional lattice. The plane it rotates sits at an irrational angle to that lattice, and the lattice's shadow on it is dense — which is why a projection needs a window before it becomes a pattern.

Where five-fold becomes legal

A five-fold rotation with whole-number entries exists — in four dimensions, as a four-by-four matrix that can be written down. The plane forbids it because the plane is too small, and knowing which dimension is large enough changes what a quasicrystal is.

restriction · Restriction Three rotations of order twelve, and one twelve-fold. Three integer 4 × 4 matrices of order twelve, one from each rational class of that order on a four-dimensional lattice. A four-dimensional rotation turns two perpendicular planes, and each plane is drawn as a dial: a point, its images, and the path the operation takes round them. Φ₁₂ turns its planes by one twelfth and five twelfths of a turn, so a point runs round twelve positions. Φ₃ ⊕ Φ₄ turns one plane by a third and the other by a quarter, and Φ₄ ⊕ Φ₆ by a quarter and a sixth; each has order twelve because twelve is the least common multiple, and neither has a plane with twelve positions in it.

Three rotations of order twelve

A four-dimensional lattice can be carried onto itself by three different integer matrices of order twelve, and only one of them turns anything by a twelfth of a turn. The totient that is usually said to decide which orders a dimension permits decides something else: which angles. The orders are decided by a sum over blocks, the two lists part company at six dimensions, and space already has an operation whose order is not its angle.

restriction · Restriction

Named alongside it

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

Cyclotomic polynomialsHigher-dimensional latticeArithmetic classCrystallographic restrictionEuler's totientHyperlatticeQuasicrystalRotoinversion

All concepts