Crystallography & Symmetry
patterns derived from their groups
Generator

A Penrose tiling, 5 inflations

A Penrose tiling, 5 inflations
A Penrose tiling, 5 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.

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.

12 essays call penrose. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. Order without repetition

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.

 One tile becomes 130 in 3 inflations. One tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell. The panel counts are 10, 20, 50, 130, and each is the one before it multiplied by the substitution matrix — an identity in whole numbers, checked every time the figure is drawn rather than quoted. Order without repetition

Inflation, and where the golden ratio comes from

A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio squared.

 Assuming a 5-fold rotation. The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist. What a lattice forbids

Why five-fold is impossible

A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.

 A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call. Order without repetition

Order is not periodicity

For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.

 The tile does not force aperiodicity — the decoration does. A rhomb with the acute angle of a Penrose tile, repeated by the lattice its own edges generate. The tiling is periodic, so the shape forbids nothing. Adding the edge decoration changes the answer: every interior edge of this tiling presents a double arrow against a single one, which the matching rule refuses. Order without repetition

Matching rules, and what actually forces aperiodicity

The two Penrose rhombs are usually said to tile the plane only aperiodically. They tile it periodically without difficulty. What cannot be done periodically is tiling them according to the decoration, and the distinction is the whole result.

 Cut and project. A square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational. Order without repetition

Cut and project

Take a periodic lattice, cut a strip through it at an irrational angle, and keep the shadow of what falls inside. The result never repeats, has exactly two spacings, and is a quasicrystal — built from something perfectly periodic that is simply not where anybody was looking.

 A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call. Order without repetition

What Shechtman measured

A diffraction pattern with sharp spots and tenfold symmetry, in April 1982. Sharpness meant order and tenfold meant no lattice, and the two had been believed inseparable — so the interesting question is what the observation had to rule out before it could mean anything.

 Which inflation factors a tiling may have. Every distinct inflation factor produced by a two-letter substitution whose matrix has entries up to 4, plotted against its algebraic conjugate. The two grey lines are the unit circle, which in the quadratic case is the pair of values ±1. A factor whose conjugate lies strictly inside is a Pisot number and the chain it grows has sharp Bragg peaks; 39 of the 77 factors here lie outside and cannot. The golden ratio is the smallest of them all, which is the arithmetic reason it turns up in every quasicrystal anybody has drawn. Order without repetition

Which inflation factors exist

A tiling grown by substitution has an inflation factor, and it is an eigenvalue of an integer matrix — so it is an algebraic integer, and sharp diffraction demands that its conjugates be small. That condition is an inequality between two integers, and it explains why the golden ratio turns up in every quasicrystal anybody has drawn.

 A quasilattice down a 5-fold axis: 10-fold, from an axis of order 5. 153 points of a three-dimensional quasilattice, made by keeping the points of Z⁶ whose perpendicular image lies inside a window and projecting them into ordinary space, then viewed along one of its 5-fold axes. There is no lattice here and no unit cell, and the symmetry is nevertheless exact: all sixty rotations of the icosahedral group carry the set onto itself, on 153 points of its core, measured by applying them rather than assumed from the construction. Seen down this axis the set comes back to itself under a turn of a 10th and no finer turn, measured over every turn up to a twelfth — twice the order of the axis, and the factor of two is a centre of symmetry: this set equals its own negation, which is what a window centred on the origin produces, and that is checked here rather than assumed. A diffraction experiment would record the same 10 whatever: the measured intensity acquires a centre whether or not the structure has one, so the tenfold photograph of 1982 does not by itself distinguish a structure with a centre from one without. Order without repetition

Six integers, and the lattice that holds them

A fivefold rotation is not an integer matrix in three dimensions and is one in six. The icosahedral group permutes its own six fivefold axes, so in coordinates along those axes every one of its sixty rotations is a signed permutation — and Z⁶ is a lattice it maps onto itself.

 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. Order without repetition

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.

 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound. Order without repetition

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

 66 squares and 106 rhombs. The Ammann–Beenker tiling, built by keeping the points of a four-dimensional lattice whose companion image falls inside an octagon and projecting them into the plane. Every tile has the same edge length; the squares and the forty-five degree rhombs are told apart by their diagonals. Nothing was placed — the faces were found among the projected points. Order without repetition

Eight-fold, with the golden ratio taken out

Every quasicrystal on this site has been built on five: Penrose's rhombs, the Fibonacci chain, the ten-fold pattern Shechtman measured. A method that works only on the golden ratio is a method tuned to its answer — so here is the same construction run on eight, where the irrational is √2 and nothing else changes.

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