Definition:Penrose Tiles
Definition
The Penrose tiles are two tiles whose rules of combination make them such that they cannot make a periodic tiling, but can make aperiodic tilings:
They both have:
- $2$ sides of length $1$
- $2$ sides of length $\phi = \dfrac {1 + \sqrt 5} 2$ (the golden mean)
The vertices are marked as shown.
To form a Penrose tiling, the vertices must match.
Penrose Kite
The Penrose kite is the larger of the two Penrose tiles.
Three of its internal angles are of $72 \degrees$, and the internal angle is of $144 \degrees$.
The $144 \degrees$ vertex and the opposite $72 \degrees$ vertex are marked to differentiate them from the other two vertices, and so as to match the $36 \degrees$ vertices of the Penrose dart.
Penrose Dart
The Penrose dart is the smaller of the two Penrose tiles.
Two of its opposite internal angles are of $36 \degrees$, one is of $72 \degrees$, and the other is of $216 \degrees$.
The $72 \degrees$ vertex and the opposite $216 \degrees$ vertex are marked to differentiate them from the other two vertices, and so as to match the opposite $72 \degrees$ vertices of the Penrose kite.
Also see
- Results about the Penrose tiles can be found here.
Source of Name
This entry was named for Roger Penrose.
Historical Note
The Penrose tiles were invented by Roger Penrose in $1974$.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Penrose tiles