Crystallography & Symmetry
patterns derived from their groups
Concept

Penrose tilings — where it appears

A tiling of the plane by two rhombs with matching rules that force it never to repeat. It has the fivefold symmetry a lattice forbids, obtained by giving up the lattice, and its tiles point in ten directions and no more.

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

A Penrose tiling, 4 inflations. Two rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself.

Penrose tilings

Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.

aperiodic · Aperiodic One patch of radius 2. A Penrose patch of 476 vertices, with the vertices within 2 edge lengths of one of them marked and the circle drawn. That marked set, written in coordinates relative to its centre, is what the census compares: two vertices have the same patch when their marked sets agree. Every vertex of the tiling is the centre of one such patch, and the question is how many different ones there are.

How many patches of each size

A periodic tiling has one kind of neighbourhood however far out you look. Random points have as many kinds as neighbourhoods. A Penrose tiling has a number in between that never stops growing and never catches up — and the count is a measurement rather than a theorem.

aperiodic · Complexity

Named alongside it

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

AperiodicityRepetitivityDiscretenessFactor complexityGolden ratioLocal indistinguishabilityLong-range orderMatching rulesMeasurementSelf similaritySturmian

All concepts