Penrose Tiles

Author

Eric W. Weisstein
August 4, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Tilings/PenroseTiles.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PenroseTiles.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Based on the example and notebook by
Lyman P. Hurd
May 19, 1994
http://library.wolfram.com/infocenter/MathSource/595/

Penrose Tiles

<<ExtraPackages`PenroseTiles`

GraphicsRow[{LineGraph1/@{Kite,Dart}}]

Deflation

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.

GraphicsGrid[{LineGraph1/@{AcuteTriangle,ObtuseTriangle}},Alignment->Bottom]

Deflation

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

GraphicsGrid[{LineGraph1/@Deflate/@{AcuteTriangle,ObtuseTriangle}},Alignment->Bottom]

Example

LineGraph1[Deflate[Sun,4]]

More Examples

Their are two ways to tile around a vertex maintaining 5-fold symmetry: sun and 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.

GraphicsGrid[{LineGraph1/@{Star,Sun}}]

GraphicsGrid[{LineGraph1[Deflate[#,3]]&/@{Star,Sun}}]

GraphicsGrid[{ColorGraph[Deflate[#,3]]&/@{Star,Sun}}]