Girih tiles are a set of five
tiles that were used in the creation of tiling patterns for decoration of buildings in Islamic architecture. They are known to have been used since about the year 1200 and their arrangements found significant improvement starting with the Darb-i Imamshrine in Isfahanin Iran built in 1453.
The five shapes of the tiles are:
* a regular
decagonwith ten interior angles of 144°;
* an elongated (irregular convex)
hexagonwith interior angles of 72°, 144°, 144°, 72°, 144°, 144°;
bow tie(non-convex hexagon) with interior angles of 72°, 72°, 216°, 72°, 72°, 216°;
rhombuswith interior angles of 72°, 108°, 72°, 108°; and
* a regular
pentagonwith five interior angles of 108°.
All sides of these figures have the same length; and all their angles are multiples of 36° (π/5). All of them, except the pentagon, have bilateral (reflection) symmetry through two perpendicular lines. Some have additional symmetries. Specifically, the decagon has tenfold rotational symmetry (rotation by 36°); and the pentagon has fivefold rotational symmetry (rotation by 72°).
Girih are lines (
strapwork) which decorate the tiles. In most cases, only the girih (and other minor decorations like flowers) are visible rather than the boundaries of the tiles themselves. The girih are piece-wise straight lines which cross the boundaries of the tiles at the center of an edge at 54° (3π/10) to the edge. Two intersecting girih cross each edge of a tile. Most tiles have a unique pattern of girih inside the tile which are continuous and follow the symmetry of the tile. However, the decagon has two possible girih patterns one of which has only fivefold rather than tenfold rotational symmetry.
Periodic or aperiodic?
Most uses of girih tiles in Islamic architecture were periodic; they had
unit cells which were repeated in the same orientation within a lattice. Some had patterns which could not be extended to a tiling of the entire plane. None of them are known to have had patterns which could be extended to the entire plane only in an aperiodic way.
However, on some buildings, the large girih tiles were decorated with patterns which formed small girih tiles. And on one of these,
Darb-i Imam, the subdivision into smaller tiles was done in a way that could have been generalized to an aperiodic tilingof the plane.
Mathematics of girih tilings
2007, Peter J. Lu of Harvard Universityand Professor Paul J. Steinhardt of Princeton Universitypublished a paper in the journal "Science" suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystallinetilings such as Penrose tilings (presentation 1974, predecessor works starting in about 1964) predating them by five centuries. [cite journal
author = Peter J. Lu and Paul J. Steinhardt
year = 2007
title = Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture
journal = Science
volume = 315
pages = 1106–1110
url = http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007.pdf
doi = 10.1126/science.1135491] [Supplemental figures [http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007_SOM.pdf] ]
This finding was supported both by analysis of patterns on surviving structures, and by examination of
15th centuryPersian scrolls. If correct, it would indicate that Islamic architects came close to discovering aperiodic tilings some five hundred years before they were discovered by Western mathematicians. Although, we have no indication of how much more the architects may have known about the mathematics involved.
* [http://www.quadibloc.com/math/pen05.htm John Savard's reconstructions]
Wikimedia Foundation. 2010.
Look at other dictionaries:
Mosaic — This article is about a decorative art. For other uses, see Mosaic (disambiguation). Irano Roman floor mosaic detail from the palace of Shapur I at Bishapur … Wikipedia
Inventions in medieval Islam — A significant number of inventions were developed in the medieval Islamic world, a geopolitical region that has at various times extended from Al Andalus and Africa in the west to the Indian subcontinent and Malay Archipelago in the east.… … Wikipedia
Quasicrystal — Atomic model of an aluminum palladium manganese (Al Pd Mn) quasicrystal surface. Similar to Fig. 6 in Ref. A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can… … Wikipedia
Tessellation — A tessellation of pavement A honeycomb is an example of a t … Wikipedia
Zellige — ( ar. الزليج) (also Zellidj, Zillij, Zellij) is terra cotta tilework covered with enamel in the form of chips set into plaster. [L Opinion (May 6 1992)] It is one of the main characteristics of Moroccan architecture though it s also used in other … Wikipedia
History of geometry — Geometry (Greek γεωμετρία ; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre modern mathematics, the other being the study of numbers. Classic geometry… … Wikipedia
Peter Lu — Peter James Lu, PhD (陸述義, b. 1978 in Cleveland, OH) is a post doctoral research fellow in the Department of Physics at Harvard University, Cambridge, Massachusetts. His most widely known discovery, amidst pursuits in several diverse fields (see… … Wikipedia
Darb-e Imam — The shrine of Darb e Imam, located in the Dardasht quarter of Isfahan, Iran, is a funerary complex, with a cemetery, shrine structures, and courtyards belonging to different construction periods and styles. The first structures were built by… … Wikipedia
List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… … Wikipedia
Paul Steinhardt — Infobox Scientist box width = 300px name = Paul J. Steinhardt image width = 250px caption = At the 2008 World Science Festival in New York City birth date = birth place = death date = death place = residence = U.S. citizenship = nationality =… … Wikipedia