In crystallography the icosahedron is a possible form, but it has not been observed; it is closely simulated by a combination of the octahedron and pentagonal dodecahedron, which has twenty triangular faces, but only eight are equilateral, the remaining twelve being isosceles (see Crystallography).

It is self-reciprocal; the cube and octahedron, the dodecahedron and icosahedron, the small stellated dodecahedron and great dodecahedron, and the great stellated dodecahedron and great icosahedron are examples of reciprocals.

Svo - Kat- rpoieKovra, thirty-two), is a 32-faced solid, formed by truncating the vertices of an icosahedron so that the original faces become triangles.

View more

The great icosahedron is the reciprocal of the great stellated dodecahedron.

The great dodecahedron is determined by the intersections of the twelve planes which intersect the Platonic icosahedron in five of its edges; or each face has the same boundaries as the basal sides of five covertical faces of the icosahedron.

The truncated icosahedron is formed similarly to the icosidodecahedron, but the truncation is only carried far enough to leave the original faces hexagons.

The truncated dodecahedron is formed by truncating the vertices of a dodecahedron parallel to the faces of the coaxial icosahedron so as to leave the former decagons.

Nevertheless, holding that every dimension has a principle of its own, he rejected the derivation of the elemental solids - pyramid, octahedron, icosahedron and cube - from triangular surfaces, and in so far approximated to atomism.

It is enclosed by 20 triangular faces belonging to the icosahedron and 12 decagons belonging to the dodecahedron.

Philolaus, connecting these ideas, held that the elementary nature of bodies depends on their form, and assigned the tetrahedron to fire, the octahedron to air, the icosahedron to water, and the cube to earth; the dodecahedron he assigned to a fifth element, aether, or, as some think, to the universe (see Plut.