Probably Plato notices this as the only remaining regular polyhedron,
which from its approximation to a globe, and possibly because, as
Plutarch remarks, it is composed of 12 x 30 = 360 scalene triangles
(Platon. Quaest.), representing thus the signs and degrees of the
Zodiac, as well as the months and days of the year, God may be said to
have 'used in the delineation of the universe.' According to Plato
earth was composed of cubes, fire of regular pyramids, air of regular
octahedrons, water of regular icosahedrons. The stability of the last
three increases with the number of their sides.

The elements are supposed to pass into one another, but we must remember
that these transformations are not the transformations of real solids,
but of imaginary geometrical figures; in other words, we are composing
and decomposing the faces of substances and not the substances
themselves--it is a house of cards which we are pulling to pieces and
putting together again (compare however Laws). Yet perhaps Plato may
regard these sides or faces as only the forms which are impressed on
pre-existent matter. It is remarkable that he should speak of each of
these solids as a possible world in itself, though upon the whole
he inclines to the opinion that they form one world and not five.
To suppose that there is an infinite number of worlds, as Democritus
(Hippolyt. Ref. Haer. I.) had said, would be, as he satirically
observes, 'the characteristic of a very indefinite and ignorant mind.'

The twenty triangular faces of an icosahedron form the faces or sides of
two regular octahedrons and of a regular pyramid (20 = 8 x 2 + 4); and
therefore, according to Plato, a particle of water when decomposed is
supposed to give two particles of air and one of fire. So because an
octahedron gives the sides of two pyramids (8 = 4 x 2), a particle of
air is resolved into two particles of fire.

The transformation is effected by the superior power or number of the
conquering elements. The manner of the change is (1) a separation of
portions of the elements from the masses in which they are collected;
(2) a resolution of them into their original triangles; and (3) a
reunion of them in new forms. Plato himself proposes the question,
Why does motion continue at all when the elements are settled in their
places? He answers that although the force of attraction is continually
drawing similar elements to the same spot, still the revolution of the
universe exerci