(4) On the inscribing of each of the five regular polyhedra in a sphere.
The general theory of polyhedra properly belongs to combinatorial analysis.
Of lost works by Archimedes we can identify the following: (I) investigations on polyhedra mentioned by Pappus; (2) Archai, Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) Peri zygon, On balances or levers; (4) Kentrobarika, On centres of gravity; (5) Katoptrika, an optical work from which Theon of Alexandria quotes a remark about refraction; (6) Ephodion, a Method, mentioned by Suidas; (7) Peri sphairopeoia, On Sphere-making, in which Archimedes explained the construction of the sphere which he made to imitate the motions of the sun, the moon and the five planets in the heavens.
He shows how to inscribe the five regular polyhedra within it.
Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.
Stevinus was the first to show how to model regular and semiregular polyhedra by delineating their frames in a plane.
If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel.
" Regular polyhedra " are such as have their faces all equal regular polygons, and all their solid angles equal; the term is usually restricted to the five forms in which the centre is singly enclosed, viz.
The Platonic solids, while the four polyhedra in which the centre is multiply enclosed are referred to as the Kepler-Poinsot solids, Kepler having discovered three, while Poinsot discovered the fourth.
Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them.
" A The following Table gives the values of A, V, R, r for the five Polyhedra: - Kepler-Poinsot Polyhedra.
A connexion between the number of faces, vertices and edges of regular polyhedra was discovered by Euler, and the result, which assumes the form E + 2' = F ± V, where E, F, V are the number of edges, faces and vertices, is known as Euler's theorem on polyhedra.
The following table gives these constants for the regular polyhedra; n denotes the number of sides to a face, n 1 the number of faces to a vertex: - Archimedean Solids.
The interrelations of the polyhedra enumerated above are considerably elucidated by the introduction of the following terms: (1) Correspondence.
Two polyhedra correspond when the radii vectores from their centres to the mid-point of the edges, centre of the faces, and to the vertices, can be brought into coincidence.
Two polyhedra are reciprocal when the faces and vertices of one correspond to the vertices and faces of the other.
Hemihedral forms are of special importance in crystallography, to which article the reader is referred for a fuller explanation of these and other modifications of polyhedra (tetartohedral, enantiotropic, &c.).
The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which one polyhedron may be derived from another.
The determination of the number of different polyhedra of n faces, i.e.