POLYGONAL NUMBERS. See NUMBER. POLYHEDRON (from Ok. HoXtledIpos, lnolyrrlros, having ninny bases, from roXin, polys, much, many + gSpa, hr(lra, base). A solid whose bounding surface consists entirely of planes. The polygons which bound it are called its !dims; the sides of those polygons, its edges; and the points where the edges meet, its vertices. If a polyhedron is such that no straight line ran lie drawn to cut its surface more than twice, it is said to be conics; otherwise it is said to lie coneare. Unless the contrary is stated the word polyhedron means convex polyhedron. If the faces of a polyhedron are congruent and regular polygons, and the polyhedral angles are all congruent, the poly hedron is said to be regular. A polyhedron which has for bases any two polygons in parallel having S faces, 6 vertices, and 12 edges, the equation becomes 12 + 2 = 8 + G. For every polyhedron there is another which, with the same number of edges, has as faces as the first has vertices, and as manv vertices as the first has faces. There cannot be more than five regular convex polyhedra. These solids are repre sented by the accompanying figures, and are sometimes known as the Platonic bodies, from the attention they received among the Platonists.
For these five polyhedra, if s be the number of sides in each face, a the number of plane angles at each vertex, then, following the other notation above given, sf =n•r = 2c. Also the
sum of all the plane angles in each figure is 2r (e-2). These formulas may easily be verified from the following table of elements: The five regular polyhedra can be constructed from cardboard by marking out the following, cutting through the heavy lines and half through the dotted ones, bringing the edges together, planes, and for lateral faces triangles or trape zoids which have one side in common with one base and the opposite vertex or side in common with the other base, is called a prismatoid. (See MENSUBAT1ON. ) In accordance with the defini tion, also all prisms and pyramids (q.v.) are in cluded among the prismatoids. Among the general relations of polyhedra, the following are the most remarkable: If a convex polyhedron has c edges, Ti vertices, and f faces, then e 2 = f r. (A theorem known to Descartes, but bearing Euler's name.) E.g.. in a regular octahedron, a solid Consult: Rouehe et Ca mberousse, Traite de Geometric (Paris, 7th ed., 1900), Eberhard, Zur Morphologic der Polgeder (Leipzig, (1S91) ; Kirkman. "On the Theory of the Polyedra," in the Philosophical Trans actions of the Royal Rociety (London, 1562, 152) ; Zeising, "Die reguliiren Polyeder," in Deutsche Vicrteljahrsschrift (Stuttgart, 18 pt. 4).