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By Root 659 0

save them, they constructed a new altar where the sides of the cube were double the length of those of the original altar. Yet by doubling the side of a cube, the volume of the cube increases by two cubed, or eight. Apollo was not happy and made the pestilence worse. The challenge that the god set – that is, given a cube, construct a second cube that has twice the volume – is called the Delian Problem, and it is one of the three classic problems of antiquity that cannot be solved by Euclidean tools. The other two are the squaring of the circle, which is the construction of a square that has the same area as a given circle, and the trisection of an angle, which is the construction of an angle that is a third of a given angle. Realizing why Euclidean geometry cannot solve these problems – and why other methods can – has been a long-time preoccupation of mathematics.

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.

The beauty of a repeating mosaic pattern lies not so much in the aesthetic appeal of the replicated image as in the effortlessness with which the tiles perfectly fill the space. The better the geometry, the better the art. Working out what shapes will tile a wall so that there are no gaps and no overlaps is quite a mathematical challenge, familiar to anyone who has ever tiled a bathroom floor. It turns out that only three of the regular polygons are able to ‘tessellate’, which is the technical word for covering a plane so that no region is uncovered. These are the equilateral triangle, the square and the hexagon. In fact, a triangle is not required to be equilateral in order to tessellate. The sides can be of any size. For any triangle, all you need to do is join it to an identical triangle placed upside-down, as in the diagram below. The combined shape is a parallelogram. The parallelogram can be joined with identical ones to form a row, and these rows can fit together side by side. This type of tessellation – in which the same pattern repeats endlessly – is called periodic.

Triangle and quadrilateral tessellations.

A square tile will fill a flat surface. That’s obvious. So does any rectangle. That is also a trivial observation – staring at a brick wall is equally the scrutiny of a at’ion of rectangles. What is surprising, however, is that any shape with four sides can also produce periodic tessellations. Draw any four-sided shape. Join this shape with one upside-down, as we did with the triangle above, and you create a six-sided shape, an irregular hexagon, with the property that each of the opposite edges are equal. Since the opposite edges are equal, the shape can be laid in a row such that the edges fit perfectly alongside each other. As the diagram on the previous page shows, this fit works in the direction of each of the sides, and the repeated hexagons cover a plane