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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.
The chakana (Andean cross, "stepped cross" or "step motif" or "stepped motif") is a stepped cross motif used by the Inca and pre-incan Andean societies. The most commonly used variation of this symbol today is made up of an equal-armed cross indicating the cardinal points of the compass and a superimposed square.
Girih patterns can be created in a variety of ways, including the traditional straightedge and compass construction; the construction of a grid of polygons; and the use of a set of girih tiles with lines drawn on them: the lines form the pattern. Patterns may be elaborated by the use of two levels of design, as at the 1453 Darb-e Imam shrine.
The union of all edges of a Cairo tiling is the same as the union of two tilings of the plane by hexagons.Each hexagon of one tiling surrounds two vertices of the other tiling, and is divided by the hexagons of the other tiling into four of the pentagons in the Cairo tiling. [4]
Covering a flat surface ("the plane") with some pattern of geometric shapes ("tiles"), with no overlaps or gaps, is called a tiling. The most familiar tilings, such as covering a floor with squares meeting edge-to-edge, are examples of periodic tilings. If a square tiling is shifted by the width of a tile, parallel to the sides of the tile, the ...
The case for a connection with Islamic mathematics is much stronger for the development of the geometric patterns with which arabesques are often combined in art. Geometric decoration often uses patterns that are made up of straight lines and regular angles that somewhat resemble curvilinear arabesque patterns; the extent to which these too are ...
Islamic geometric patterns are derived from simpler designs used in earlier cultures: Greek, Roman, and Sasanian. They are one of three forms of Islamic decoration , the others being the arabesque based on curving and branching plant forms, and Islamic calligraphy ; all three are frequently used together.
Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids.The well-known toy version was developed by British engineer Denys Fisher and first sold in 1965.