Penrose Tiles

Implementation

Needs["PenroseTiles`"]

Example

The following command gives the illustration at the top of this 
notebook.

LineGraph1[Deflate[Sun,4]]

Penrose Tiles

Roger Penrose discovered a The original Penrose tiles consist of two tiles which can tile the plane but cannot do so in a periodic fashion. The two tiles are the Kite:

LineGraph1[Kite]

And the Dart:

LineGraph1[Dart]

The tiles must be placed in such a way that the colored markings agree;  in particular, the two tiles may not be combined into a rhombus.

We divide the kite into two acute isoceles triangles (angles 72, 72, 36 degrees) and the dart into two obtuse triangles (angles 36, 36, 108 degrees).  The reason for this decomposition will be more apparent below when we discuss the operation Penrose calls deflation.

LineGraph1[AcuteTriangle]

LineGraph1[ObtuseTriangle]

Internally we represnt a tiling by a list of triangles of the form a[x,y,z], and o[x,y,z] respectively.  In each case x,y, and z, are planar coordinates of the form {r1,r2}, machine-precision real numbers.

The Deflation Operator

Deflation and inflation are operators on legal tilings.  The deflation operator applied to a finite collection of tiles, leads to a more refined collection.  However, as usually defined, the operators do not respect tile boundaries.  The operators do, however, respect half tiles.  This fact is the reason for defining tilings in terms of triangles rather than tiles.

The "deflation" operator takes an acute triangle to the union of two acute triangles and one obtuse.

LineGraph1[Deflate[AcuteTriangle]]

The obtuse triangle goes to an acute and an obtuse.

LineGraph1[Deflate[ObtuseTriangle]]

Note that in each case the LineGraph[] routine only draws two sides of each triangle.  This convention makes pairs of triangles form tiles.

More Examples

Their are two ways to tile around a vertex maintaining 5-fold symmetry.  Penrose calls these configurations Sun, Star.  Deflation maintains this symmetry and in the limit gives the two possible 5-fold symmetric Penrose tilings.  These two possibilities are taken into each other by deflation.

LineGraph1[Sun]

LineGraph1[Star]

LineGraph1[Deflate[Sun,3]]

ColorGraph[Deflate[Sun,4]]

LineGraph1[Deflate[Sun,4]]

Bibliography

B. Grunbaum & G. C. Shepard. Tilings and Patterns (W. H. Freeman and Company: New York, 1987).

Credits

By Lyman P. Hurd

This package is similar to one shown to the author in 1988 by 
William Thurston.

Copyright 1994 Lyman P. Hurd