(4) On the inscribing of each of the five regular polyhedra in a sphere.
The general theory of polyhedra properly belongs to combinatorial analysis.
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 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.