Crystallography & Symmetry
patterns derived from their groups
Concept

Triangulation — where it appears

A division of a polygon into triangles by diagonals that do not cross. A polygon with n corners has Catalan-many of them, and each one gives a closed frieze of numbers whose second row counts the triangles at every corner.

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

A frieze of numbers built by one rule, and the glide it was never given. The frieze pattern of order 7 with second row 1, 3, 2, 1, 4, 1, 3: rows of ones at top and bottom, and every diamond b, a, d, c of neighbouring entries satisfying ad − bc = 1. The pattern repeats every 7 steps, and it is also carried onto itself by a glide reflection that turns it upside down and moves it 3.5 steps; the two tinted blocks are one block and its image under the glide. Its frieze group is p11g. The heptagon beside it is the triangulation whose triangle counts at the vertices give the second row.

A rule with no mirror hands every strip a glide

Fill a strip with numbers by one local rule: every diamond of four neighbours b, a, d, c has ad − bc = 1. The rule is read the same way everywhere and contains no reflection. Yet every such strip that closes, with a row of ones returning, is carried onto itself by a glide reflection that turns it upside down. So of the seven frieze groups only four can occur, and which of them a strip has is the symmetry of a triangulated polygon, halved and turned. Of 23,712 strips up to order twelve, not one has a group without a reflection along its length.

classification · Friezes A frieze of numbers that never closes. The frieze pattern with top row of ones and second row 1, 2, 6 repeated, every diamond b, a, d, c obeying ad − bc = 1, drawn to its 8th row. Every entry is a positive integer and the rows grow without end: the matrix that carries the pattern through one period has trace 3, above 2, so no row of ones returns and there is no bottom edge. With no bottom edge, nothing can turn the strip over, and the only symmetries left are translations and mirrors across it; this one has none of the mirrors, and its frieze group is p1.

Friezes of numbers find six groups and never p2

A frieze of numbers whose row of ones never returns is an infinite strip with one edge, and with one edge nothing can turn it over. Such friezes are abundant — at trace two there are C(2n − 1, n) of each period, where closing gave Catalan numbers — and every one is p1 or p1m1. The p1 ones come in left- and right-handed pairs, which no closed frieze can. Together with the four groups closed friezes are forced into, that is six of the seven, and the seventh, p2, is ruled out by a three-step argument that uses the glide.

classification · Friezes

Named alongside it

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

Frieze groupGlide reflectionLocal rulesChiralityClassificationEnantiomorphForcingStrip

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