In the Timaeus Plato gives the icosahedron to water. The reasoning follows the pattern of the other assignments: among the solids built from triangular faces it is the largest and the most nearly round, so it rolls and flows most easily, and it is matched to the element that flows. Fire gets the sharpest solid, earth the steadiest, air the one in between, and water the one that moves without cutting.
The scheme is more than a table of pairs. Because fire, air and water are all built from the same elementary triangle, they can be broken apart and reassembled into one another, and the dialogue gives the accounting: the twenty triangular faces of a particle of water can be redistributed as two particles of air, at eight faces each, and one of fire, at four. That is a mechanism for change of state, offered with numbers, in the fourth century BC.
It should still be read for what it is. Plato is not describing water molecules, which are not icosahedral and are not made of triangles, and he offers no evidence beyond the fitness of the analogy. The passage is a philosopher's reconstruction of what an intelligible universe would have to be like. Nearly every subsequent association of the icosahedron with water, emotion, flow and adaptability traces back to it, and where a modern source presents that link as ancient universal wisdom, the honest description is that it comes from one dialogue by one author.
What survives the correction is the ambition. Plato asked why substances behave differently and answered in terms of structure rather than of quality, which is the right kind of question to ask. The answers were wrong. The form of the question was not.