Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay

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Some of the first modern work on tilings was done by Johannes Kepler. He also was one of the first mathematicians to treat star polygons as regular polygons. This tiling is based on one of Kepler's figures. None of the component polygons tile the plane periodically. Is this a true aperiodic tiling? |

No, it isn't. Although none of the
individual polygons tile the plane periodically, the
entire set does. The repeat polygon is a rhombus with
vertices in the centers of the blue paired decagons. Nevertheless, Penrose Tilings (below) were inspired by Kepler's Tilings. |

Kepler also discovered a number of ways pentagons and related polygons can cover portions of the plane. |

Roger Penrose discovered a number of aperiodic tilings with pentagonal symmetry. This one bears an obvious kinship to Kepler's tilings. This version, showing the shapes of the six component tiles, also tiles periodically. | |

The tiles above can be made aperiodic by modifying some of the points, but they can also be marked as shown here. The rule is that colors have to match across all adjacent edges. |

In 1974, Penrose discovered an aperiodic tiling that uses only two shapes, nicknamed
kites and darts. When people use the term "Penrose
tiling", this is usually what they mean. The tiles
can be prevented from tiling periodically by putting
notches and tabs on the edges of the tiles, but a more
aesthetic approach is to color the tiles as shown and
require the edges to match. This pattern, named for the star at the center, is called the Infinite Star Pattern. |

There are an infinity of Penrose Tilings.
This one contains a decagon at its center instead of a
star (It's called the Infinite Sun Pattern). Yet note
that large parts of the two patterns are similar. Penrose Tilings have a number of astonishing properties. One is that any finite portion of any tiling is contained an infinite number of times in every other tiling. This means that you cannot tell, by examining a piece of a tiling, which pattern you are on! |

In some ways the pattern here, the
Cartwheel, is the most important Penrose Tiling. The
purple region at the center is outlined by a decagon
consisting of a kite and dart edge. Every point in every
tiling is contained inside an identical decagon (although
the contents may differ). The outer portion of the pattern consists of two parts. There are ten yellow sectors and ten blue spokes. The spokes consist of "bowtie" units and the spokes can be flipped 180 degrees and still fit their adjacent sectors. That means there are 1024 possible spoke arrangements but after rotations and reflections are eliminated there are only 62 distinct patterns. |

Each of the spokes can be continued inside the cartwheel, but eventually they end, enclosing a region that Penrose calls a "decapod" (in red). All decapods can be constructed out of 36-108-36 isoceles triangles. The one shown here, dubbed "Batman", is the only one of the 62 decapods that can be legally tiled with darts and kites. The five darts in a Batman figure and the two intervening kites are the only tiles in any Penrose tiling that are ever part of a pattern without fivefold symmetry.

Penrose tilings can also be based on rhombi. The acute angles in the rhombi are 36 and 72 degrees. Coloring the rhombi as shown forces aperiodicity. |

Rhombus tiles can be generated from dart and kite tiles and vice versa. The Infinite Sun and Star Patterns above are shown in purple, and the corresponding rhombi in black. |

The most remarkable property of Penrose Tilings is that *every
finite portion of any tiling is contained infinitely often in
every other tiling.* This, of course, is true of all periodic
tilings, but it's not at all obvious that it should be true of a
non-periodic tiling. This property has several consequences:

- No finite patch of tiles can force a tiling (determine the rest of the tiling).
- It is impossible to tell from any patch of tile which tiling it is on.
- Only at their infinite limits are the different patterns distinguishable. A finite patch of an Infinite Star pattern might only be a local piece of some other pattern, but there is also an Infinite Star pattern that has five-fold symmetry to infinity. Only if you know the characteristics of the pattern to infinity can you tell.

This property is both less and more remarkable than it seems.
For example, consider the numbers pi (3.14159265358979326433....),
*e* (2.718281828459045235360287...) and the square root of
2 (1.4142135623730950488...). All of them contain the number
sequences ..23.. and ..35.. . In fact, it is widely believed (though not
formally proven) that any finite sequence of
digits will be contained infinitely often in all three numbers,
and no finite sequence of digits will enable you to tell which
number you are looking at (except, of course, for the integer and
decimal point).

What's remarkable about the Penrose Tilings is how dense the
patterns are. The sequence ...89793... occurs infinitely often in
pi, *e*, and the square root of 2, but only every 100,000
digits on the average, and the actual spacing could be vastly
greater. In fact, there is no known upper limit. If a patch of
tiles in a Penrose tiling has a diameter d, there will be an
identical patch within a distance of at most 2d and most likely
within d. (See how close to the center of the cartwheel above you
can find another Batman.)

Martin Gardner, Extraordinary nonperiodic tiling that enriches
the theory of tiles, Mathematical Games, *Scientific American*,
January, 1977, p. 110-121

Grunbaum, B and Shephard, G. C., *Tilings and Patterns*,
Freeman, 1987. Just about everything there was to know on the
subject at the time.

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*Created 11 August 1999, Last Update 11 August 1999*

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