Frieze Groups, Wallpaper Groups, and Girih Tilings

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 Library
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