How to use Polyhedra in a sentence
Mensuration of the Platonic Solids.-The mensuration of the regular polyhedra is readily investigated by the methods of elementary geometry, the property that these solids may be inscribed in and circumscribed to concentric spheres being especially useful.
The module can be varied to produce a large number of related rhombic polyhedra and polyhedral combinations.
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.
Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them.Advertisement
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.
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.Advertisement
The determination of the number of different polyhedra of n faces, i.e.
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.