484 



ON CONCRETIONS, SPICULES, ETC. 



CH. 



Within any polyhedron we may always inscribe another 

 polyhedron, Avhose corners correspond in number to the sides or 

 facets of the original figure, or (in alternative cases) to a certain 

 number of these sides ; and a similar result is obtained by bevelling 

 ofi the corners of the original polyhedron. We may obtain a 

 prec'isely similar symmetrical result if (in such a case as these 

 Radiolarians which we are describing), we imagine the radial 

 spines to be interconnected by tangential rods, instead of by the 

 complete facets which we have just been dealing with. In our 

 complicated polyhedron with its twenty radial spines arranged in 

 the manner described there are various symmetrical ways in which 

 we may imagine these interconnecting bars to be arranged. The 

 most symmetrical of these is one in which the whole surface is 

 divided into eighteen rhomboidal areas, obtained by systematically 

 connecting each group of four adjacent radii. This figure has 

 eighteen faces (F), twenty corners (C), and therefore thirty-six 

 edges (E), in conformity with Euler's theorem, F + C = E + 2. 



Another symmetrical arrange- 

 ment will divide the surface 

 into fourteen rhombs and eight 

 triangles. This latter arrange- 

 ment is obtained by linking up 

 the radial rods as follows : aaaa, 

 aba, abcb, bcdc, etc. Here we 

 have again twenty corners, but 

 we have twenty-two faces; the 

 number of edges, or tangential 

 spicular bars, will be found, 

 therefore, by the above formula, 

 to be forty. In Haeckel's figure 

 of Phractaspis prototypus we 

 have a spicular skeleton which 

 appears to be constructed precisely upon this plan, and to 

 be derivable from the faceted polyhedron precisely after this 

 manner. 



In all these latter cases it is the arrangement of the axial 

 rods, or in other words the "polar symmetry" of the entire 

 organism, which lies at the root of the matter, and which, if only 



Fig. 235. Phractaspis prototypus, Hkl. 



