Penrose Tiling

By Joshua John Clark, Daniel Kerstiens, Jason Piercy, and Caleb Rouleau

Outline:

Overview of Penrose Tiling Example: Artist: Urs Schmid Photo by: Urs Schmid Date: drawn in 1995

Who is Roger Penrose?

 1. Oxford professor
 2. PhD from Cambridge

note: penrose is from 500 years before him (check wikipedia)

Definition of non-periodic

(http://en.wikipedia.org/wiki/Non-periodic)

Types of Penrose Tilings

  1. Original Pentagonal
  2. Kite and Dart
  3. Rhombus

Penrose Tilings have two different types of symmetry: reflectional and rotational. A tiling's style of symmetry overall is referred to as local pentagonal symmetry[1]. As these tilings are non-periodic, however, they do not have translational symmetry.

A penrose tiling can't have more than one point of global five-fold symmetry. The reason for this fact is that rotating about an extra point of global five-fold symmetry would generate two closer centers of five-fold symmetry, which causes a contradiction[2].

Also, the golden ratio (1 + sqrt(5)) / 2 appears within several aspects of Penrose tilings[3].

Rules for construction

  1. Matching rules
  2. Substitution tiling

Other tilings

References

[1] Austin, David (2005), "Penrose Tiles Talk Across Miles", Feature Column (Providence: American Mathematical Society).

[2] Gardner, Martin (1997), Penrose Tiles to Trapdoor Ciphers, Cambridge University Press, ISBN 978-0-88385-521-8.

[3] Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Gallian, J. (2013). Contemporary abstract algebra. (8th ed.). Boston, MA: Brooks/Cole, Cengage Learning. http://www.ics.uci.edu/~eppstein/junkyard/tiling http://www.ics.uci.edu/~eppstein/junkyard/penrose.html