<P> Islamic art makes use of geometric patterns and symmetries in many of its art forms, notably in girih tilings . These are formed using a set of five tile shapes, namely a regular decagon, an elongated hexagon, a bow tie, a rhombus, and a regular pentagon . All the sides of these tiles have the same length; and all their angles are multiples of 36 ° (π / 5 radians), offering fivefold and tenfold symmetries . The tiles are decorated with strapwork lines (girih), generally more visible than the tile boundaries . In 2007, the physicists Peter Lu and Paul Steinhardt argued that girih from the 15th century resembled quasicrystalline Penrose tilings . Elaborate geometric zellige tilework is a distinctive element in Moroccan architecture . Muqarnas vaults are three - dimensional but were designed in two dimensions with drawings of geometrical cells . </P> <P> Ibn Muʿādh al - Jayyānī is one of several Islamic mathematicians to whom the law of sines is attributed; he wrote his The Book of Unknown Arcs of a Sphere in the 11th century . This formula relates the lengths of the sides of any triangle, rather than only right triangles, to the sines of its angles . According to the law, </P> <Dl> <Dd> sin ⁡ A a = sin ⁡ B b = sin ⁡ C c . (\ displaystyle (\ frac (\ sin A) (a)) \, = \, (\ frac (\ sin B) (b)) \, = \, (\ frac (\ sin C) (c)).) </Dd> </Dl> <Dd> sin ⁡ A a = sin ⁡ B b = sin ⁡ C c . (\ displaystyle (\ frac (\ sin A) (a)) \, = \, (\ frac (\ sin B) (b)) \, = \, (\ frac (\ sin C) (c)).) </Dd>

How were muslim states governed in the early middle ages