How do you
tile a plane in such a way that its
aperiodic? This means
that although there may be local
symmetries, the overall pattern has no
specific patch is ever
repeated. This is in contrast to
classic square
and
triangular tile systems that have repeatable patches as they cover the
infinite plane.
The Penrose tiles were discovered in 1974 by Roger Penrose. Later, in
1984, it was demonstrated that when these tiles were fit together using a simple rule set, they will cover an infinite plane in an infinite number of arrangements without ever repeating.
There are two types of Penrose tiles: Dart & Kite, and Rhombs. Both of these use decoration rules for determining where a tile can be placed. Unfortunately, I can't reproduce either set without mangling the angles which are important.
Because of the difficulty explaining the Dart & Kite, I shall focus on the Rhombs. The narrow rhomb has angles of 144o and 36o. The thick rhomb has angles of 108o and 72o. Alone, these two tiles can be placed in periodic patterns, it is the tiles and the rules that make up Penrose tiles. The obtuse angle on the narrow rhombs are both decorated differently with matching decorations on the acute angles on the thick rhomb. When placing tiles, the decorations must match up.
It has been proven that Penrose tiles can be colored with three colors such that no two adjacent tiles have these same color.