114 // Miles of Tiles

continent, in the middle of which is an enormous lake. Pff lives on an island in the middle of the lake. Neither Nff nor Pff can swim, fly or teleport: their only form of transport is to walk on dry land. Yet each morning, one walks to the other's house for breakfast. Explain.

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Answers on page 277

Miles of Tiles Bathroom walls and kitchen floors provide everyday examples of tiling patterns, using real tiles, plastic or ceramic. The simplest pattern is made from identical square tiles, fitted together like the squares of a chessboard. Over the centuries, mathematicians and artists have discovered many beautiful tilings, and mathe- maticians have gone a stage further by seeking all possible tilings with particular features.

For instance, exactly three regular polygons tile the entire infinite plane ­ that is, identical tiles of that shape cover the plane without overlaps or gaps. These polygons are the equilateral triangle, the square and the hexagon:

The three regular polygons that tile the plane.

We can be confident that no other regular polygon tiles the plane, by thinking about the angles at which the edges of the tiles meet. If several tiles meet at a given point, the angles involved must add to 3608. So the angle at the corner of a tile is 3608 divided by a whole number, say 360/m. As m gets larger, this angle gets smaller. In contrast, as the number of sides of a regular polygon increases, the angle at each corner gets bigger. The effect Miles of Tiles // 115

of this is to `sandwich' m within very narrow limits, and this in turn restricts the possible polygons.

The details go like this. When m ¼ 1, 2, 3, 4, 5, 6, 7, and so on, 360/m takes the values 360, 180, 120, 90, 72, 60, 513, and so on.

7 The angle at the corner of a regular n-gon, for n = 3, 4, 5, 6, 7, and so on, is 60, 90, 108, 120, 1284, and so on. The only places where

7 these lists coincide are when m ¼ 3, 4, and 6; here n ¼ 6, 4 and 3.

Actually, this proof as stated has a subtle flaw. What have I forgotten to say?

The most striking omission from my list is the regular pentagon, which does not tile the plane. If you try to fit regular pentagonal tiles together, they don't fit. When three of them meet at a common point, the total angle is 361088 ¼ 3248, less than 3608. But if you try to make four of them meet, the total angle is 461088 ¼ 4328, which is too big.

Irregular pentagons can tile the plane, and so can innumer- able other shapes. In fact, 14 distinct types of convex pentagon are known to tile the plane. It is probable, but not yet proved, that there are no others. You can find all 14 patterns at www.mathpuzzle.com/tilepent.html mathworld.wolfram.com/PentagonTiling.html

The mathematics of tilings has important applications in crystallography, where it governs how the atoms in a crystal can be arranged, and what symmetries can occur. In particular, crystallographers know that the possible rotational symmetries of a regular lattice of atoms is tightly constrained. There are 2-fold, 3-fold, 4-fold and 6-fold symmetries ­ meaning that the arrangement of atoms looks identical if the whole thing is rotated through 1, 1, 1 or 1 of a full turn (3608). However,

2 3 4 6 5-fold symmetries are impossible ­ just as the regular pentagon cannot tile the plane.

There the matter stood until 1972, when Roger Penrose discovered a new type of tiling, using two types of tile, which he called kites and darts: 116 // Miles of Tiles

A kite (left) and dart (right). The matching rules require the thick

and thin arcs to meet at any join ­ see the pictures below.

These shapes are derived from the regular pentagon, and the associated tilings are required to obey certain `matching rules' where several tiles meet, to avoid simple repetitive patterns. Under these conditions, the two shapes can tile the plane, but not by forming a repetitive lattice pattern. Instead, they form a bewildering variety of complicated patterns. Precisely two of these, called the star and sun patterns, have exact fivefold rotational symmetry.

The two fivefold-symmetric Penrose tilings. Left: star pattern;

right: sun pattern. Tinted lines illustrate the matching rules.

Black lines are edges of tiles.

It then turned out that nature knows this trick. Some chemical compounds can form `quasicrystals' using Penrose patterns for their atoms. These forms of matter are not regular

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