More+on+Maths

"[|Girih] patterns constitute a wide-ranging decorative idiom throughout Islamic art and architecture. Previous studies of medieval Islamic documents describing applications of mathematics in architecture suggest that these girih patterns were constructed by drafting directly a network of zigzagging lines (sometimes called strapwork) with the use of a compass and straightedge. The visual impact of these girih patterns is typically enhanced by rotational symmetry. However, periodic patterns created by the repetition of a single "unit cell" motif can have only a limited set of rotational symmetries, which western mathematicians first proved rigorously in the 19th century C.E.: Only two-fold, three-fold, four-fold, and six-fold rotational symmetries are allowed. In particular, five-fold and 10-fold symmetries are expressly forbidden. Thus, although pentagonal and decagonal motifs appear frequently in Islamic architectural tilings, they typically adorn a unit cell repeated in a pattern with crystallographically allowed symmetry. Although simple periodic girih patterns incorporating decagonal motifs can be constructed using a "direct strapwork method" with a straightedge and a compass (as illustrated in [|Fig. 1, A to D]), far more complex decagonal patterns also occur in medieval Islamic architecture. These complex patterns can have unit cells containing hundreds of decagons and may repeat the same decagonal motifs on several length scales. Individually placing and drafting hundreds of such decagons with straightedge and compass would have been both exceedingly cumbersome and likely to accumulate geometric distortions, which are not observed." from "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture" **Peter J. Lu and Paul J. Steinhardt**