Penrose Tilings
Penrose tilings constitute a class of non-periodic tilings of the plane. A tiling of the plane, as the name suggests, is a covering of the entire plane by shapes (tiles), no two of which overlap. A tiling can have almost any imaginable form, but the most interesting and most carefully studied tilings are those in which all of the tiles are identical copies of just a few different tiles. A tiling of this type is said to be periodic if there is a pattern that repeats itself--more precisely, if there is some small block of the tiling that, when it is shifted about by translations, will cover the entire tiling (a translation is an operation on the plane that shifts the position of every point by some fixed amount in some fixed direction). Many of the designs of the celebrated artist M. C. Escher are periodic tilings; for example, in his "Fish and birds" tiling, a single block consisting of one fish and one bird will cover the entire tiling if it is shifted about.
There are many ways to construct non-periodic tilings, but for a long time it was believed that any set of tiles that could make a non-periodic tiling could also be arranged into a periodic tiling. In 1961 Hao Wang began to study possible arrangements of colored square tiles called Wang dominoes, and in 1964 Robert Berger built on Wang's work to prove that there is a set of Wang dominoes that can only tile the plane non-periodically. Berger was able to construct such a set, but it was extremely complicated, with over 20,000 dominoes. He was later able to reduce the number to 104, and then Raphael Robinson reduced it to 24. By examining a different type of tile, Robinson was later able to find a set of just 6 tiles that can only tile non-periodically.
The mathematician and physicist Roger Penrose set out to improve on these results, and in 1974 he found a pair of tiles that will only form non-periodic tilings. The two tiles are very simple, formed by cutting a parallelogram into two pieces (in a carefully prescribed manner); the two pieces are called a 'kite' and a 'dart' (Penrose has also found a non-periodic tiling that uses two rhombic tiles). Copies of these tiles fit together in all sorts of interesting ways, but if certain restrictions are placed on the way the tiles can be put next to each other, or if small modifications are made to their shapes, then only non-periodic tilings can occur. Although the darts are smaller than the kites, and so one might guess that more darts than kites are needed, there are more kites than darts, in a very precise way: the ratio of kites to darts is the golden mean, 1.61803398.... The appearance of the golden mean is just one of the many ways in which Penrose tilings, despite their aperiodicity, show a remarkable degree of order. Even though they have no repeating patterns, Penrose tilings are highly pleasing to the eye.
Most of the time, if you try to lay down kites and darts so that they never overlap, you will end up with a small space that fits neither a kite nor a dart. In spite of that, there are in fact infinitely many different Penrose tilings, and Penrose and mathematician John Conway have shown that there are in fact uncountably many different Penrose tilings (that is, there are more Penrose tilings than there are whole numbers). Penrose has discovered another startling fact about his tilings: if you limit yourself to a finite piece of one of the tilings, you can always find that identical piece in every other tiling, no matter how large the piece is or what its design. In other words, there is no way to tell which Penrose tiling you have just by looking at a finite piece.
Another unusual attribute of Penrose tilings is that some of the tilings have 'five-fold symmetry'. We say that an object has n-fold symmetry if there is some center of rotation for which the object remains the same if you rotate it by a 1/n-turn. For example, the square has 4-fold symmetry, since a 1/4-turn about the center will leave the square unchanged; a regular hexagon has 6-fold symmetry, since a 1/6-turn about the center leaves it unchanged. Geometers realized long ago that any periodic tiling of the plane can only have 2-fold, 3-fold, 4-fold or 6-fold symmetries; 5-fold symmetry, and 7-fold symmetry or higher, are strictly forbidden. This is not so easy to prove, but is motivated by the fact that the only regular polygons that tile the entire plane are the equilateral triangle, the square, and the regular hexagon; pentagons, septagons, and other regular polygons cannot fill the entire plane without overlaps or gaps. Thus, mathematicians were fascinated when Penrose showed that in the realm of non-periodic tilings, 5-fold symmetry can occur.
Penrose tilings at first appear to belong purely to the domain of recreational mathematics; however, they have recently proven very important in the field of crystallography. When atoms form crystals, they are arranged into periodic tilings of space. For a long time, crystallographers believed that the only two possible types of solids were crystals (periodic tilings) and glasses (completely random arrangements of atoms). When Penrose discovered his tilings, scientists began to ask whether there was anything in nature corresponding to this new geometric structure. In 1982, Dan Shechtman discovered an alloy that displayed the forbidden five-fold symmetry; his alloy and subsequently discovered alloys have been given the name 'quasicrystals'. The atomic structure of these quasicrystals is not yet known, but scientists have found some evidence supporting the idea that they may be 3-dimensional analogues of Penrose's tilings.
Although Penrose tilings don't seem like material for legal disputes, they found their way into the courts when Penrose, who patented his tilings, sued the Kimberley-Clark Corporation for embossing his design on rolls of Kleenex Quilted bathroom tissue. "So often we read of very large companies riding rough-shod over small businesses or individuals," said David Bradley, the director of Pentaplex, which has the exclusive rights to license the tilings. "But when it comes to population of Britain being invited by a multi-national to wipe their bottoms on the work of a Knight of the Realm, then a last stand must be made." Beyond politics of globalization and the farcical nature of this case lies an important question: should mathematicians 'own' their discoveries, or do mathematical discoveries belong to everyone?
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