Religion
Islam
Below: Some Islamic flavored patterns generated semi-algorithmically by computer.
There are many varieties of pattern designs by different cultures. The decorative Celtic knots and Islamic geometric tiling are particularly mathematical. Although there are great number of existing designs throughout the world and history, but with respect to mathematical or algorithmic classifications, there are not many varieties. This is not surprising when we consider that these artistic patterns are developed culturally by artisans, not as a result of mathematical analysis of plane symmetry or systematic algorithmic exploration.
Some of the Islamic Tessellations can be categorizes under type P6M:
Explanation & Background
Repeating shape is the regular hexagon.
Inside the hexagon there is a rotation over
60, 120, 180, 240 and 300
degrees, the whole horizontally mirrored.
System P6M is very suitable to design symmetric patterns.
Celtic
Above: a traditional Celtic pattern (by Alastair Luke). The main attraction of Celtic knot patterns is their geometric design, but the topological aspect (the "knot") is also interesting mathematically. One can survey traditional designs to see whether they are really knots, or braids of how many strands, and how they are knotted. In so far as I know, this has not been studied much. There is little systematic study of algorithmic knot/weaving-pattern generations. (There has been systematic mathematical studies of weavings/fabrics by B Grunbaum and G C Shephard).