An experiment in 3D repetitive tiling

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 Imam shrine in Isfahan in Iran built in 1453. The five shapes of the tiles are:

• a regular decagon with ten interior angles of 144°;

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

• a bow tie (non-convex hexagon) with interior angles of 72°, 72°, 216°, 72°, 72°, 216°;

• a rhombus with interior angles of 72°, 108°, 72°, 108°; and

• a regular pentagon 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. (from Wikipedia)

An article about the 2D geometry of girih (Science magazine)

An illustrated supplement to Science magazine article

A Girih fantasy on YouTube

My 3D experiment