# Girih tiles

**Girih tiles**are a set of fivetile s that were used in the creation of tiling patterns for decoration of buildings inIslamic architecture . They are known to have been used since about the year 1200 and their arrangements found significant improvement starting with theDarb-i Imam shrine inIsfahan in Iran built in 1453.The five shapes of the tiles are:

* a regulardecagon with ten interior angles of 144°;

* an elongated (irregular convex)hexagon with interior angles of 72°, 144°, 144°, 72°, 144°, 144°;

* abow tie (non-convex hexagon) with interior angles of 72°, 72°, 216°, 72°, 72°, 216°;

* arhombus with interior angles of 72°, 108°, 72°, 108°; and

* a regularpentagon with 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 cell s 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 anaperiodic tiling of the plane.**Mathematics of girih tilings**In

2007 , Peter J. Lu ofHarvard University and Professor Paul J. Steinhardt ofPrinceton University published a paper in the journal "Science" suggesting that girih tilings possessed properties consistent withself-similar fractal quasicrystalline tilings such asPenrose tiling s (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 century Persian 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.**See also***

Zellige **References****External links*** [

*http://www.quadibloc.com/math/pen05.htm John Savard's reconstructions*]

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*Wikimedia Foundation.
2010.*

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