POLYGONACE^. 218 POLYHEDRON. amphibium, one of the species of the section or suborfler Persicaria, is abundant ahont margins of ponds and ditches throughout the Northern HeniispheW. It has two forms of leaves; those npoM the erect stems being broad and smooth, those wliieh (loat in the water narrow and rough, difTcrenees wliieh might be held to indicate dis- tinct species, yet both may be found growing from one root. The stems have been used on the C'ontinent of Euroiw as a substitute for sar- saparilla. Several species arc occasionally used for dyeing, as the spotted persicaria ( Polygonum Persicaria), a very common weed on manure heaps and in waste places in Kurope and also naturalized in the United States. The only spe- cies really important on this account is that called dyers' buckwheat (Polygonum tincto- rium). a Chinese biennial, with ovate leaves and slender spikes of reddish flowers. It has been successfully cultivated in France and Flanders. It j'ields a blue dye scarcely inferior to indigo. Coceoloba. another genus of this order, has a wide distribution throughout the tropics, Coccoloha iivifcra and other species producing edible fruits. POLYGONAL NTTMBEBS. See Number. POLYHEDRON (from (ik. UoUeSpos, pol- yeilrus, having many bases, from ttoXus, polys, much, many + ISpa, hrdra, base). A solid whose bounding surface consists entirely of planes. The polygons which bound it are called its faces; 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 can he dra«Ti to cut its surface more than twice, it is said to be convex; otherwise it is said to he ronriirr. Unless the contrary is stated t)ie 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 8 faces, G vertices, and 12 edges, the eqiation becomes 12 + 2 = 8 -f (!. For ever}' polyhedron there is another which, with the same numlier of edges, has as many faces as the first has vertices, and as many 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 kno^Mi 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, n the number of plane angles at each vertex, then, following the other notation above given, s/^ = «.• r = 2c. Also the sum of all the plane angles in each figure is 2 ir ( i> — 2). These formulas may easily be verified from the following table of elements: POLYHEDRA. 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 Mensueation.) In accordance w-ith 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 e edges. V 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 NAME OF 80MD s n 1 V e Tetrahedron 3 3 4 4 6 Hexahpdron 4 3 6 8 12 Octahedron 3 4 8 6 12 Dodecahedron... 5 3 12 2U SO Icoaahedron » 5 20 12 30 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. IC08ABEDB0K. TETBIBEDBOK. BEXAHEDBON. DODECAHEDRON. Consult : Rouch§ et Cam berousse, Traite de G^ometrie ( Paris, 7th ed., innO), Eberhard. Zttir Morphologic der Polyeder ( Leipzig, (1891); Kirkman. "On the Theory of the Polyedra," in the Philosophical Trans- actions of the Royal S^ociety (London. 1862. vol. 152) ; Zeising. "Die reguliiren Polyeder." in the Deutsche Vierteljahrsschrift (Stuttgart, 1869, pt. 4). OCTAHEDRON.
