Frieze Groups, Wallpaper Groups, and Girih Tilings

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Frieze Groups, Wallpaper Groups, and Girih Tilings

Abdhullah Khan Madrasa, Uzbekistan (1588) photo by Peter Lu www.peterlu.org photo by Peter Lu www.peterlu.org A frieze pattern is a 2-dimensional pattern with exactly one direction of translational symmetry and with the property that there is a translational symmetry of smallest length. A wallpaper pattern is a 2-dimensional pattern with two directions of translational symmetry and with the property that there is a translational symmetry of smallest length. '&% f(x)=x/2, %&$ g(x)=x +1 %&# %&" %&! 1 Not a frieze b/c h(x)=x + 2n ! is a symmetry" # for every$ n. %'& &'$ &'# g(x)=x +1 &'" &'! is a frieze ! " # $ %& Theorem: There are exactly 7 symmetry types of frieze patterns. Alhambra photo by Brian Sanderson http://www.math.toronto.edu/drorbn/Gallery/Symmetry/Tilings/ photo by Dror Bar-Natan http://www.math.toronto.edu/drorbn/Gallery/Symmetry/Tilings/ photo by Dror Bar-Natan http://www.math.toronto.edu/drorbn/Gallery/Symmetry/Tilings/ Yeşil Cami (1421) Yeşil Cami (1421) http://en.wikipedia.org/wiki/Yeşil_Mosque#mediaviewer/File:Bursa014.jpg Theorem: There are exactly 17 symmetry types of wallpaper patterns. wikipedia Girih Tiles (Peter Lu) archnet.org/sites/1639 Harvard College LibraryHarvard Gunbad-i Qabud, Maragheh, Iran (1197) Girih tiles highlighted by Peter Lu http://www.boston.com/news/science/articles/2007/02/26/science_imitates_art/ scroll in Tokapi Palace, Istanbul http://www.boston.com/news/science/articles/2007/02/26/science_imitates_art/ Kepler’s 11 Archimedean tilings (David Austin) http://www.ams.org/samplings/feature-column/fcarc-penrose An aperiodic tiling A Penrose Tiling http://en.wikipedia.org/wiki/Penrose_tiling#mediaviewer/File:Penrose_Tiling_(Rhombi).svg archnet.org/sites/3901 Darb-i-Imam Shrine, Iran (1453) For more: • Lu and Steinhardt “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture.” Science 315 (2007): 1106-1110. • Peter Lu in the Harvard Physics Seminar at peterlu.org • Austin “Penrose tiles talk across miles” http://www.ams.org/samplings/feature-column/fcarc-penrose.

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Medieval Islamic Architecture, Quasicrystals, and Penrose and Girih Tiles: Questions from the Classroom
Medieval Islamic Architecture, Quasicrystals, and Penrose and Girih Tiles: Questions from the Classroom Raymond Tennant Professor of Mathematics Zayed University P.O. Box 4783 Abu Dhabi, United Arab Emirates [email protected] Abstract Tiling Theory studies how one might cover the plane with various shapes. Medieval Islamic artisans developed intricate geometric tilings to decorate their mosques, mausoleums, and shrines. Some of these patterns, called girih tilings, first appeared in the 12th Century AD. Recent investigations show these medieval tilings contain symmetries similar to those found in aperiodic Penrose tilings first investigated in the West in the 1970’s. These intriguing discoveries may suggest that the mathematical understanding of these artisans was much deeper than originally thought. Connections like these, made across the centuries, provide a wonderful opportunity for students to discover the beauty of Islamic architecture in a mathematical and historical context. This paper describes several geometric constructions for Islamic tilings for use in the classroom along with projects involving girih tiles. Open questions, observations, and conjectures raised in seminars across the United Arab Emirates are described including what the medieval artisans may have known as well as how girih tiles might have been used as tools in the actual construction of intricate patterns. 1. Islamic Tilings and Traditional Strapwork The Islamic world has a rich heritage of incorporating geometry in the construction of intricate designs that appear on architecture and tile walkways as well as patterns on fabric, see [4]. This highly stylized form of art has evolved over the centuries from simple designs to fairly complex geometry involving a high degree of mathematical symmetry.
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