32. Lamprias: The matter is thus explained by Theodorus of Soli:[[128]] There are five and no more solid figures having all the faces and all the solid angles in each equal. These are—
(a) The Pyramid (Tetrahedron) with four faces, each an equilateral triangle, and four solid angles,
(b) The Cube, six faces, each a square, and eight solid angles,
(c) The Octahedron, eight faces, each an equilateral triangle, and six solid angles,
(d) The Dodecahedron, twelve faces, each a regular pentagon, and twenty solid angles,
(e) The Eicosahedron, twenty faces, each an equilateral triangle, and twelve solid angles.
[It follows that (d) having more, and blunter, solid angles than any, most nearly approximates to the Sphere. (And, in fact, if the content of the Sphere be 100, that of (d) is 66·5, that of (e) only 60·5, that of (c) 36·75, and so on). Plato (Timaeus, pp. 53-5, where see Archer-Hind) shows that each equilateral triangle may easily be broken into six ‘primary scalenes’, i. e. triangles with angles 90°, 60°, 30°, which again will reproduce themselves ad infinitum (Euclid, 6, 8). Hence, if a universe be constructed out of (a) or (c) or (e) or their plane faces, or of all of these, it can, in case of dissolution, be reconstructed. This does not apply to the Cube, the faces of which, however, yield isosceles right-angled triangles, also available as ‘constituents’ in infinite number, nor yet to (d) which is therefore reserved for another purpose, as to which see Burnet (Early Greek Philosophers, c. 7, sect. 148).]
The solid figures may be used to construct five different worlds, or omitting (d) for the four ‘elements’ (fire, &c.).
33. Ammonius criticizes; he points out that the difficulty about the figure (d) has been ignored.
34. Lamprias drops the subject for the present, and turns to the five categories of being in the Sophistes and Philebus. It is reasonable to assume that the physical universe may correspond.