In 1973 Roger Penrose found a set of six tiles that force aperiodicity.
Later, in 1974 he was able to reduce the set to two tiles. The shape of
a pair of Penrose tiles can vary but the most common are the kite and
the dart. These tiles are derived from a rhombus with angles 72 and 108
degrees.
Kite and Dart
There are different matching rules that enforce a way of joining the
tiles in such a way as to create an aperiodic tiling. One of them
consists of using two colors to label the corners and forcing fitting
edges's corners to be of the same color. Another rule consists of
drawing circular arcs of two colors on each side and edges must join
arcs of the same color in order to join.
Matching rules for the
kite and dart
Another set of tiles can be derived by bisecting the kite and dart into
isosceles triangles and merging the triangles into thick and thin rhombs.
Penrose rhombs
For example, a portion of a Penrose tile using the rhombs looks like
this:
Penrose tiling
Generation of Penrose tilings
There are many ways of generating a Penrose tiling. The most obvious
one is to make many copies of the tiles and start arranging them
according to the matching rules. Using this approach one might reach a
state where no more matchings are possible and we need to backtrack to
make different choices for the position of the tiles.
Penrose tilings can also be generated using a substitution method. On
every tile we can draw smaller generations of the tiles and similarly
from smaller tiles we can generate larger tiles. This also means that
there is a infinite number of Penrose tilings because we can continue
substituting any number of times. The penrose tiling above was generated
using this method with the postcript Penrose tiler described later.
A more involved method which lets us generate a Penrose tiling and to
look at any portion of it consists of projecting a portion of a
5-dimensional lattice onto a 2-dimensional plane. By translating the
plane we can draw any part of the Penrose tiling.
Postscript Penrose tiling generator
The Postscript Penrose tiling generator can be found here: penrose.ps
This postscript program uses the substitution method to draw Penrose
tilings. It uses a set of 4 tiriangular tiles that are generated by
bisecting the thin rhomb along its short diagonal and the thick rhomb
along its long diagonal. We will call these triangles TL, TR, tL and tR
as shown below. The arrows on the sides specify the matching rules.
Triangular tiles
Each of these triangles can be substituted by smaller triangles. The
substitutions are:
Triangle substitutions
The program starts with a set of triangles and performs a substitution
on each of the triangles. It does the same for the new tiling and so on
for a specified amount of times. By using different starting sets and
varying the number of iterations it can generate a large set of tilings,
only limited by the postscript interpreter's stack size.
The next image shows the tilings generated starting with one TL and one
TR triangle and different number of iterations.
Multiple iterations
To set the starting triangle set open the penrose.ps file and change
the value of the triangles
variable. The value is an array of triangles and each triangle has the
following format:
[type direction base]
type is 0 if it is a T triangle or 1 if it is a t triangle
direction if 0 for an L triangle or 1 for an R triangle
base consists of two arrays that represent the endpoints of the
triangle's base: [x1 y1] [x2 y2].
For example, the value for triangles
used in the image above was:
[[0 0 [3 8] [5 5]] [0 1 [5 5] [3 8]]]
The value of the variable N controls the number of iterations the
program will perform.
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Penrose tilings of
the plane
Generation of Penrose tilings
