Octahedron

The octahedron has eight triangular faces, six vertices and twelve edges, with four faces meeting at each vertex. Euler's formula gives 6 - 12 + 8 = 2. The easiest way to see the shape is as two square pyramids joined base to base, and that hidden square is one of three mutually perpendicular squares that can be cut through the solid.

It is the dual of the cube, and the relationship is worth stating precisely rather than waving at. The cube has six faces and eight vertices; the octahedron has eight faces and six vertices; both have twelve edges. Join the centres of the cube's six square faces and an octahedron appears inside it. Join the centres of the octahedron's eight triangular faces and a cube appears inside that. The counts swap, the edges stay put, and the two solids share a single symmetry group of 48 operations, of which 24 are rotations. This is why a cube and an octahedron are, from the point of view of symmetry, one object seen from two sides.

Its dihedral angle, the angle between two faces along a shared edge, is about 109.47 degrees, more exactly the angle whose cosine is minus one third. That figure reappears in chemistry as the bond angle of a tetrahedral molecule, and the repetition is not coincidence: it is the angle subtended at the centre of a regular tetrahedron by two of its corners, and the same geometry underlies both.

Among the five, the octahedron sits in the middle almost everywhere. It has more faces than the tetrahedron and fewer than the icosahedron. More faces meet at its vertices than at the cube's, fewer than at the icosahedron's. Its dihedral angle falls between theirs. It is also the one of the five that most readily balances on a point, since its opposite vertices are directly aligned. All of that is genuine geometry, and it is why the shape reads as balance to anyone building a symbolic vocabulary out of the solids. The symbolic reading itself is a modern habit rather than a classical one.

In the Timaeus Plato gives the octahedron to air, and the reason is positional. Three of his elements are built from solids with triangular faces: fire takes the tetrahedron, air the octahedron, water the icosahedron. They are assigned in order of size and mobility, so air, standing between fire and water in both, takes the middle solid. Once again the argument is one of fitness, and what it records is Plato's reasoning, not a property of the atmosphere.

Because those three solids have faces built from the same elementary triangle, Plato allows them to transform into one another, and he is willing to do the arithmetic. The faces can be redistributed: an icosahedron carries twenty triangles, an octahedron eight and a tetrahedron four, and twenty is two eights and a four. The dialogue accordingly describes a particle of water breaking up into two of air and one of fire. Whatever else this is, it is a mechanism, offered with numbers attached, and that is unusual for its date.

Air in the ancient scheme also carried the sense of breath, and the association of the octahedron with breath and with the throat in later writing draws on that older layer rather than on anything geometric. It should not be read back into the Timaeus, which is interested in mobility and cutting edges rather than in respiration.

The whole construction is best understood as an early and serious attempt at what would now be called a theory of matter: elements with structures, structures with parts, and rules by which the parts recombine. It is wrong in every particular. It is not foolish, and it is not mysticism, and the difference matters when weighing what later writers made of it.

The octahedron's real-world life is chemical, and it is more interesting than its symbolism. When a metal ion is surrounded by six other atoms or molecules, they arrange themselves at the corners of an octahedron, because that spreads six things as far apart as possible around a centre. Octahedral coordination is the commonest arrangement in the chemistry of the transition metals, and it governs the structure, the reactivity and the colour of an enormous range of compounds. It is not a decorative fact: the geometry determines how the electron energy levels split, and that in turn determines what the substance looks like and how it behaves.

Crystals follow suit. Fluorite, magnetite, spinel and diamond all grow in octahedral habits, and diamond octahedra are common enough to be a recognised form in the rough. As with cubic salt, the outward shape is the internal lattice made visible, and the two habits are related exactly as the two solids are. Cubic and octahedral crystal forms belong to the same symmetry class, which is called the cubic system precisely because it contains both.

In close-packed structures the gaps between spheres come in two kinds, and the larger of them is called the octahedral hole because it is bounded by six spheres arranged as an octahedron. A great deal of solid-state chemistry is bookkeeping about which ions sit in which holes.

Modern esoteric systems assign the octahedron to the element air, to the throat chakra and to the east. Those attributions are modern, drawn from twentieth-century New Age and crystal literature, and only the air link goes back to Plato. The chakra assignment does not appear in classical Indian sources, where the chakras are figured as flat yantras: the air centre is conventionally shown as a hexagram within a circle, not as a solid. Different modern authors also place the octahedron at different chakras, which is a reliable sign of an invented rather than a transmitted correspondence.