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.