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.