Hexahedron (Cube)

The cube, or hexahedron, has six square faces, eight vertices and twelve edges, with three faces meeting at each vertex. Euler's formula holds as it must for a convex polyhedron: 8 - 12 + 6 = 2. The angle between two adjacent faces is exactly 90 degrees, and the cube is the only one of the five for which that angle is a whole number of degrees.

It is the dual of the octahedron, and the pairing is exact rather than approximate. The cube has six faces and eight vertices; the octahedron has eight faces and six vertices. Mark the centre of each square face of a cube, join the neighbours, and an octahedron appears inside it. Do the same to the octahedron's eight triangles and a cube comes back. The edge count of twelve is unchanged in both directions, and the two solids share one symmetry group of 48 operations, 24 of them rotations. Any statement about the symmetry of a cube is a statement about the symmetry of an octahedron.

The cube's other distinction is practical rather than ornamental. It is the only regular solid that tiles three-dimensional space: identical cubes stacked face to face fill a volume with no gaps and no overlaps, and none of the other four can do it. Aristotle believed regular tetrahedra packed perfectly as well, and he was wrong; the mistake was repeated for a very long time before it was properly corrected. That single tiling property is why the cube, rather than the more elegant icosahedron, became the shape of the built world. A good deal of what gets written about the cube as a symbol of order and foundation is downstream of a fact about packing.

One caution about that symbolism. The cube is stable in the sense that it sits flat and stacks, but it is not rigid. A cube built from rods and hinges racks over into a leaning box, while a tetrahedron built the same way cannot deform at all. Stability and rigidity are different properties, and the cube has only the first of them.

In the Timaeus Plato assigns the cube to earth. The reason he gives is steadiness: earth is the least mobile of the elements, and of the five solids the cube has the most secure bases, sitting flat on whichever face it is set down. As with the other assignments this is an argument from fitness rather than from evidence, and it is a fact about what Plato wrote rather than a fact about soil.

The cube occupies a special position in his scheme. Plato builds the faces of the solids out of two elementary right triangles. The tetrahedron, octahedron and icosahedron all have triangular faces made from the half-equilateral triangle, and because they share a building block they can be dismantled and recombined into one another: fire, air and water pass back and forth. The cube alone is built from the half-square, and Plato states plainly that earth therefore cannot be turned into any other element. Break it up as you like and it reassembles as earth.

That is an unusually specific piece of theorising, and it is the nearest thing in the passage to a testable proposal. It is also wrong, which is worth saying without embarrassment. Matter is not composed of nested right triangles, and the four elements of the ancient scheme are not elements in the modern sense at all. What makes the Timaeus interesting here is not that it reached the right answer but that it attempted a geometrical account of substance, roughly two thousand years before there was any means of testing one.

Two attributions are commonly overstated. The four-element scheme is not Plato's invention; it is usually traced to Empedocles in the fifth century BC, and Plato inherited it. The five solids are not Plato's discovery either. What the dialogue contributed was the marriage of the two: elements given shapes, and shapes given a mechanism.

Cubic crystals are real, common and well understood. Sodium chloride, ordinary table salt, grows in cubes because its sodium and chloride ions alternate on a cubic lattice, and the flat outer faces are that internal arrangement showing through at visible scale. Galena, fluorite and pyrite do the same. The insight that outward crystal form follows from a repeating internal structure is the foundation of crystallography, and it is a better story than any symbolic reading of the shape.

In scripture the cube appears as a deliberate figure for completeness. The inner sanctuary of Solomon's temple is described in the first book of Kings as twenty cubits in length, width and height, a cube, and the most restricted space in the building. The New Jerusalem at the end of Revelation is measured by an angel and found equal in length, breadth and height, a city in the form of a cube on a scale that is plainly symbolic rather than architectural. In both passages the equality of the three dimensions is what carries the meaning.

One frequent claim needs correcting. The Kaaba in Mecca is often called a perfect cube, and its Arabic name does carry the sense of a cube. It is not one. The structure is a rectangular block whose base measures roughly twelve metres by ten, with unequal sides and a height greater than either. Its significance in Islam has nothing to do with its proportions, and no Islamic source treats it as a Platonic solid.

Modern esoteric systems assign the cube to the root chakra, to the north, and to Saturn. These are twentieth-century developments out of New Age and crystal-healing writing, and they are neither classical nor Indian: the traditional element diagram for earth in Indian practice is a flat yellow square, and the classical chakra system uses two-dimensional yantras rather than solids. Modern sources also disagree about which solid belongs to which chakra, which is itself good evidence that the correspondence is recent rather than inherited.