734 A NOTE ON POLYHEDRA [ch. 



that in a polyhedron of C corners, the sum of the plane 

 angles 



= 4 (C - 2) right angles (2)*. 



Hence, if the polyhedron be isogonal, the sum of the plane angles 

 at each corner 



4 (C - 2) 



C 



90°, or U - ^i right angles (3). 



The five regular sohds, or Platonic bodies — there can be no more 

 — ^have been known from remote antiquity ; they have their corners 

 all ahke and their face's all alike, they are isogonal and isohedral. 

 Three of them, the tetrahedron, octahedron and icosahedron, have 

 triangular faces; three of them, the tetrahedron, cube and dodeca- 

 hedron, have trihedral or three-way corners. One or other of these, 

 triangles or three-way corners, must (as we shall soon see) be present 

 in every polyhedron whatsoever. 



The semi-regular solids are -regular in one respect or other, but 

 not in both; they are either isogonal or isohedral — isohedral, when 

 every face is an identical polygon and isogonal when at every corner 

 the same set of faces is combined. The semi-regular isogonal gohds, 

 with all their corners ahke but with two or more kinds of regular 

 polygons for their faces, are thirteen in number — there can be no 

 more; they were all described by Archimedes, and we call them 

 by his name. One of them, with six square and eight hexagonal 

 facets, derived by truncating the octahedron or the cube, we have 

 found to be of pecuhar interest, and it has become famihar to us 

 as, of all homogeneous space-fillers, the one which encloses a given 

 volume within ,a minimal area of surface. It is the cuho-octahedron 

 of Kepler or of Fedorow, the tetrakaidekahedron of Kelvin, which 

 latter name we commonly use. 



Of semi-regular isohedral bodies, with all their sides alike (though 

 no longer regular polygons) and their corners of two kinds or more, 

 only one was known to antiquity; it is the rhombic dodecahedron, 

 which is the crystalline form of the garnet, and appears in part 



* On this remarkable parallel see Jacob Steiner, Gesammelte Werke, i, p. 97. 

 It follows that the sum of the plane angles in a polyhedron, as in a plane polygon, 

 is at once determined by the number of its comers: a result which dehghted 

 Euler, and led him to base his primary or generic classification of polyhedra on 

 their comers rather than their sides. 



