Penrose Tiling Generator

A bathroom floor repeats. Slide the whole pattern one tile to the right and it lands exactly on itself, and a single small block, copied side by side, rebuilds everything. That sliding move is called a translational symmetry, and a tiling that has one is periodic. Most people assume, without ever saying so, that any orderly tiling must work this way.

A Penrose tiling is orderly and has no such move. Every tile sits where a short list of rules allows, the same small clusters turn up again and again, and yet no shift of the plane carries the pattern onto itself. The surprise is not disorder. It is order of a kind that people had not thought possible.

A question from logic, answered with two tiles

It is easy to lay tiles without a period: scatter the joints of a brick wall and the pattern never repeats. That proves little, because the same bricks also lay in neat rows. The hard question, raised by Hao Wang in the 1960s while working on a problem in logic, was whether some set of shapes can tile the plane only without a period.

Robert Berger found the first such set within a few years. It needed more than 20,000 different tiles. Others cut the number down, and in 1973 Roger Penrose found a set of six, built around the regular pentagon. The following year he reduced it to two.

The two tiles come in two well known versions. One pair is the kite and the dart, quadrilaterals with corner angles of 72, 72, 72 and 144 degrees and of 36, 72, 36 and 216 degrees. The other pair is a thick rhomb with angles of 72 and 108 degrees and a thin rhomb with angles of 36 and 144. The versions are closely related: cut every tile along its axis into triangles, regroup the triangles, and a tiling by one pair becomes a tiling by the other.

Five-fold symmetry is the heart of it. A short argument shows that a tiling with a translational symmetry can have rotation centers of order 2, 3, 4 or 6 but never 5. Penrose’s tiles are full of 36 and 72 degree angles, so their tilings contain patches with exact five-fold symmetry, some of them very large.

The rule on the edges

Shape alone does not do the work. Two rhombs can be laid in plain rows like any parallelogram, and a kite and a dart fit together into a single rhombus that tiles periodically. So the tiles carry marks, and a mark on one tile must agree with the mark on its neighbor across every shared edge.

The marks can be drawn several ways. One uses two colors on the corners. Another, the one most often pictured, puts two circular arcs on each tile, centered on corners, and requires that arcs continue from tile to tile. With the arcs in place the forbidden rhombus cannot be assembled, because the arcs of the kite and the dart would not meet.

Follow the rule and only a few arrangements can occur around a point. For kites and darts there are exactly seven. Seven is not many, and they are known by names borrowed from playing cards and the sky: sun, star, ace, deuce, jack, queen and king. The sun is five kites around a point and the star is five darts. For the rhombs there are eight arrangements, two of which use the same five angles.

The rule has a weakness as a building method. Lay tiles one at a time, obeying it at every edge, and you will usually reach a gap that no tile can fill. The rule tells you which finished tilings are legal. It does not tell you how to get there.

Cutting tiles into smaller tiles

Subdivision does. Cut each kite and dart, or each rhomb, down its axis into two mirror-image triangles. There are two triangle shapes, one with angles of 36, 72 and 72 degrees and one with 36, 36 and 108, and each comes left-handed and right-handed. They are called Robinson triangles.

Each triangle can be cut into two or three smaller triangles of the same two shapes, by placing a new point on one or two of its sides at the golden section. Do this to every triangle of a legal patch, glue the small triangles back into tiles, and the result is a legal patch of tiles that are smaller by the golden ratio, about 1.618. The new tiles do not respect the outlines of the old ones, only the outlines of the half tiles. Repeat, and a single tile grows into a patch of any size.

The cutting also runs backward, and that is where the lack of a period comes from. In a Penrose tiling the small tiles group into larger ones in exactly one way. Suppose some shift mapped the tiling onto itself. Because the grouping is unique, the same shift would map the tiling of larger tiles onto itself, and the next one up, and so on. Tiles in that tower grow without limit, and eventually each one is far wider than the shift. No tiling can be moved onto itself by less than the width of its own tiles. So there was no shift.

Notice what that argument uses: the whole plane and an endless tower of scales. No finite patch can carry it. Every finite patch of a Penrose tiling occurs infinitely often within it, so a patch cannot even say which tiling, or which part of one, it came from. A drawing can show that the rules were kept. It cannot show aperiodicity, and anyone who says a picture proves it has mistaken the theorem for the illustration.

The golden ratio in the counts

The golden ratio is in the shapes: the long and short sides of a kite or a dart are in that ratio, and so are the areas of the two tiles in each pair. It is also in the counts, and here the arithmetic is small enough to do by hand.

Under one step of subdivision a half kite becomes two half kites and a half dart, and a half dart becomes one of each. So if a patch holds k half kites and d half darts, the next one holds 2k + d and k + d. The same two formulas govern the thick and thin half rhombs. Start from a sun, which is ten half kites, and the counts run 10 and 0, then 20 and 10, 50 and 30, 130 and 80, 340 and 210, 890 and 550.

Those are Fibonacci numbers in disguise, and their ratio closes in on 1.618 quickly: 2, 1.6667, 1.625, 1.6190, 1.6182. Each step cuts the distance from the golden ratio by a factor of about 0.146, the fourth power of its reciprocal. A ratio that settles on an irrational number is itself a sign that no finite block is being repeated, since a repeated block would fix the ratio at a fraction.

Here is a patch small enough to check. Choose kites and darts, the sun, and depth 4. The patch has 165 kites and 100 darts, 265 tiles in all, after twenty half tiles at the rim are left out. It has 301 vertices and 565 edges, and 565 is 301 + 265 - 1, as Euler’s formula requires of a patch with no hole. Every one of its interior edges is shared by two tiles whose arcs meet. Turn one tile’s arcs end for end and four edges stop matching.

Checks like these are only as good as the arithmetic under them. Coordinates built from the golden ratio are irrational, and rounded decimals will sooner or later call two copies of one corner different points. The cure is to store each corner as whole-number multiples of the fifth roots of unity, which is exact, and to use decimals only when drawing.

Lines, atoms and lawyers

In 1981 N. G. de Bruijn gave a different construction. Draw five families of parallel lines, one family for each of five directions. The tiling is the dual of that grid: each crossing of two lines stands for one rhomb, thick or thin according to the angle of the crossing. He showed that Penrose tilings can be obtained this way.

The tilings left mathematics the following year. On 8 April 1982 Dan Shechtman, on sabbatical at the National Institute of Standards and Technology, put a rapidly cooled alloy of aluminum and manganese under an electron microscope and saw a pattern with five-fold symmetry. Crystals were then defined by periodic structure, and periodic structure rules out five-fold symmetry. It took more than two years to get the result into print. Five weeks after it appeared, Paul Steinhardt and Dov Levine set it in a wider theory and gave the material its name: quasicrystal.

In 1992 the International Union of Crystallography changed its definition of a crystal to admit such structures. Shechtman received the Nobel Prize in Chemistry in 2011.

The pattern also had a legal life. Penrose applied for a British patent on 25 June 1975 and a United States patent a year later; the United States patent, “Set of tiles for covering a surface”, was granted on 9 January 1979. A patent from that period ran for the longer of seventeen years from grant or twenty from filing, and the public record lists this one as expired. In April 1997 a writ was served on Kimberly-Clark over a quilted toilet paper embossed with the pattern. The Science Museum Group, which keeps a pack of it in its collection, records that the matter was settled out of court.