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| − | + | 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]. | ||
| Line 49: | Line 50: | ||
'''References''' | '''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. | 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/tiling | ||
http://www.ics.uci.edu/~eppstein/junkyard/penrose.html | http://www.ics.uci.edu/~eppstein/junkyard/penrose.html | ||
Revision as of 16:31, 26 November 2013
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
