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.
The names of these five solids are: (r) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 20 equilateral triangles.
The first three were certainly known to the Egyptians; and it is probable that the icosahedron and dodecahedron were added by the Greeks.
The equilateral triangle is the basis of the tetrahedron, octahedron and icosahedron.'
These solids played an important part in the geometry of the Pythagoreans, and in their cosmology symbolized the five elements: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube), universe or ether (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 great icosahedron is the reciprocal of the great stellated dodecahedron.
Each of the twenty triangular faces subtend at the centre the same angle as is subtended by four whole and six half faces of the Platonic icosahedron; in other words, the solid is determined by the twenty planes which can be drawn through the vertices of the three faces contiguous to any face of a Platonic icosahedron.
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.
It is enclosed by 20 triangular faces belonging to the original icosahedron, and 12 pentagonal faces belonging to the coaxial dodecahedron.
The truncated icosahedron is formed similarly to the icosidodecahedron, but the truncation is only carried far enough to leave the original faces hexagons.
It is therefore enclosed by 20 hexagonal faces belonging to the icosahedron, and 12 pentagonal faces belonging to the coaxial dodecahedron.
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.
It is enclosed by 20 triangular faces belonging to the icosahedron and 12 decagons belonging to the dodecahedron.
- Two 62-faced solids are derived from the dodecahedron, icosahedron and the semi-regular triacontahedron.
In the " small rhombicosidodecahedron " there are 12 pentagonal faces belonging to the dodecahedron, 20 triangular faces belonging to the icosahedron and 30 square faces belonging to the triacontahedron.
The pentagons belong to a dodecahedron, and 20 triangles to an icosahedron; the remaining 60 triangles belong to no regular solid.
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.
Thus the faces of the cuboctahedron, the truncated cube, and truncated octahedron, correspond; likewise with the truncated dodecahedron, truncated icosahedron, and icosidodecahedron; and with the small and great rhombicosidodecahedra.