Four of the five solids are given elements in the Timaeus. The dodecahedron is not. Plato writes that there remained a fifth construction, and that the god used it for the whole, embroidering figures upon it, which in context means that the dodecahedron is employed for the cosmos itself rather than for anything within it. The pentagonal solid is the roundest and most sky-like of the five, and it takes the sky.
That is all he says, and the point matters because of what was later added. The dodecahedron is constantly described as Plato's aether or quintessence. Plato does not use those terms of it. A fifth element for the heavens is Aristotle's contribution, and the joining of that element to the fifth solid is a later development within the Platonic tradition, taken up in the Renaissance and repeated ever since as though it were in the dialogue. Reporting it as Plato's word is the commonest single error made about the five solids.
The most ambitious attempt to find the cosmos in them came nineteen centuries later. In Mysterium Cosmographicum of 1596 Johannes Kepler proposed that the six known planets are spaced as they are because the five regular solids are nested between their spheres: an octahedron between Mercury and Venus, an icosahedron between Venus and Earth, a dodecahedron between Earth and Mars, a tetrahedron between Mars and Jupiter, and a cube between Jupiter and Saturn. Five solids allow exactly six spheres, so the model appeared to explain why there were six planets, a question nobody else had thought to ask.
It is wrong. The fit was never good, the better observations he inherited from Tycho Brahe made it worse, and the later discovery of further planets removed its central appeal. It is also frequently misreported that Kepler abandoned it. He did not: he issued a second edition in 1621, long after his own laws of planetary motion had superseded it. The same conviction that the heavens were built on ratio is what carried him to the laws that actually worked.