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The girih are piece-wise straight lines that cross the boundaries of the tiles at the center of an edge at 54° (3π/10 radians) to the edge. Two intersecting girih cross each edge of a tile. Most tiles have a unique pattern of girih inside the tile that are continuous and follow the symmetry of the tile.
A labyrinth can be generated by tiles in the form of a white square with a black diagonal. As with the quarter-circle tiles, each such tile has two orientations. [3] The connectivity of the resulting labyrinth can be analyzed mathematically using percolation theory as bond percolation at the critical point of a diagonally-oriented grid.
Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings ...
Patterns may be elaborated by the use of two levels of design, as at the 1453 Darb-e Imam shrine. Square repeating units of known patterns can be copied as templates, and historic pattern books may have been intended for use in this way. The 15th century Topkapı Scroll explicitly shows girih patterns together with the tilings used to create them.
Middle stage patterns on geometric borders around a Mihrab in the Alâeddin Mosque, Konya, Turkey. 1220 onwards. The next development, marking the middle stage of Islamic geometric pattern usage, was of 6- and 8-point stars, which appear in 879 at the Ibn Tulun Mosque, Cairo, and then became widespread. [25]
Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices. Wang tiles: 32: E 2: 1986 [51] Locally derivable from the Penrose tiles. No image: Wang tiles: 24: E 2: 1986 [51] Locally derivable from the A2 tiling. Wang tiles: 16: E 2: 1986 [17] [52] Derived from tiling A2 and its Ammann bars. Wang tiles: 14: E 2: 1996 [53] [54 ...