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 3 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 Every vertex of a Penrose patch, by the symmetry of what lies within three edges of it. A region of a Penrose rhomb tiling 15.2 edges across. Each vertex is marked by the local group of the patch of radius 3.05 edges about it: a short line along the mirror where that patch has one (98 vertices), a large dot where it has a five-fold centre (7), and a small dot where it has no symmetry at all (118). The mirrors are common and point in the tiling's ten directions; the five-fold centres are rare; nothing has a half-turn.

Five-fold centres stay few, and mirrors do not

The Fibonacci chain has one palindrome of every even length and two of every odd length, for ever. Run the same census on the patches of a Penrose tiling and the plane gives three different answers at once. Patches with a five-fold centre stop at four, however large they are drawn. Patches with a mirror grow in number with their radius, by about fifty for every edge. And no patch wider than a single tile has a half-turn at all. What decides which answer an operation gets is the size of the set it leaves fixed in internal space.

aperiodic · Complexity

Named alongside it

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

AperiodicityFactor complexityMatching rulesRepetitivityCut-and-projectDiscretenessGolden ratioHalf-turnLocal indistinguishabilityLocal symmetryLong-range orderMeasurement

All concepts