736 A NOTE ON POLYHEDRA [ch. 



The derivatives of the cube (with its six sides and eight corners) 

 have the following numbers of sides : 



F^ = F+C =6 + 8 =14 



F+C + E =6 + 8+12 = 26 



i^+C+2j6;=6 + 8 + 24 = 38. 



The derivatives of the dodecahedron have, in hke manner, 32, 62 

 or 92 sides; while the tetrahedron yields, by truncation of its four 

 corners, a solid with eight sides. 



The growth and form of crystals is a subject alien to our own, 

 yet near enough to attract and tempt us. It is a curious thing 

 (probably traceable to the Index Law of the crystallographer) that 

 the Archimedean or isogonal bodies seldom occur and certainly play 

 no conspicuous part in crystallography, while several of Catalan's 

 isohedral figures are the characteristic forms of well-known minerals *. 



Just as we pass from the Platonic to the Archimedean bodies by 

 truncating the corners or edges of the former, so conversely, by 

 producing their faces to a limit we obtain another family of figures 

 — in all cases save the tetrahedron, which admits of no such 

 extension; and the figures of this family are remarkable for the 

 "twinned," or duphcate or multiple appearance which they present. 

 If we extend the faces of an octahedron we get what looks hke two 

 tetrahedra, "twinned with" or interpenetrating one another; but 

 there has been no interpenetration in the construction of this twin- 

 like figure, only further accretion upon, and extension of, the facets 

 of the octahedron. Among the higher polyhedra there are many 

 figures which look, in a far more complicated way, like the twinning 

 of simpler but still comphcated forms; and these also have been 

 constructed, not by interpenetration, but by the mere superposition 

 of new parts on old. 



An elementary, even a very elementary, knowledge of the theory 

 of polyhedra becomes useful to the naturahst in various ways. 

 Among organic structures we often find many-sided boxes (or what 

 may be regarded as such), like the capsular seed-vessels of plants, 

 the skeletons of certain Radiolaria, the shells of the Peridinia, the 

 carapace of a tortoise, and a great many more. Or we may go 



* E.g. the triakis, tetrakis and hexakis octahedra of fluor-spar. However the 

 Archimedean tetrakaidekahedron [QF^ SFq) occurs in ahmi. 



