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.