The Greeks were not the only people intrigued by the wonders of geometrical shapes. The most sacred object in Islam is a Platonic solid: the Ka’ba, or Cube, is the black palladium at the centre of Mecca’s Sacred Mosque, around which pilgrims walk anticlockwise during the Hajj. (In fact, its dimensions make it just off a perfect cube.) The Ka’ba also marks the point that worshippers must face during daily prayer, wherever they are in the world. Mathematics plays more of a role in Islam than in any other major religion. More than a millennium before the advent of GPS technology, the requirement to face Mecca relied on complicated astronomical calculations—which is one reason why Islamic science was unequalled for almost a thousand years.
Islamic art is epitomized by the ingenious geometrical mosaic arrangements on the walls, ceilings and floors of its sacred buildings, a consequence of the ban on images of people and animals in holy sites. Geometry was thought to express truth beyond what was merely human, much in keeping with the Pythagorean position that the universe reveals itself through mathematics. The symmetries and endless loops that Islamic craftsmen created in their patterns were an allegory of the Infinite, an expression of the sacred, mathematical order of the world.
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The discovery of nonperiodic tiling was an exciting breakthrough for maths, but not as exciting as it would later be for physics and chemistry. In the 1980s researchers were amazed to discover a type of crystal that they did not believe existed. The tiny structure displayed a nonperiodic pattern, behaving in three dimensions just like Penrose’s tiles did in two. The existence of these structures—called quasicrystals —changed the way scientists understood the nature of matter, since it contradicted classical theory that all crystals must have symmetrical lattices derived from the Platonic solids. Penrose may have invented his tiles for fun, but they were unduly prophetic about the natural world.
Half a millennium ago, Islamic geometers might also have understood about non-periodic tessellations. In 2007 Peter J. Lu from Harvard University and Paul J. Steinhardt from Princeton claimed that their studies of mosaics in Uzbekistan, Afghanistan, Iran, Iraq and Turkey showed that the craftsmen had made ‘nearly perfect quasi-crystalline Penrose patterns, five centuries before discovery in the West’. It is possible, therefore, that Islamic mathematics may have been even more advanced than historians of science have traditionally thought.
