724 ON CONCRETIONS, SPICULES, ETC. [ch. 



symmetries and the apparent simplicity of these latter forms makes 

 of them a class apart, and suggests problems which have not been 

 solved or even investigated. 



The matter is partially illustrated by the accompanying figures 

 (Fig. 340) from Haeckel's Monograph of the Challenger Radiolaria*. 

 In one of these we see a regular octahedron, in another a regular, 

 or pentagonal, dodecahedron, in a third a regular icosahedron. In 

 all cases the figure appears to be perfectly symmetrical, though 

 neither the triangular facets of the octahedron and icosahedron, 

 nor the pentagonal facets of the dodecahedron, are necessarily plane 

 surfaces. In all of these cases, the radial spicules correspond to the 

 comers of the figure; and they are, accordingly, six in number 

 in the octahedron, twenty in the dodecahedron, and twelve in the 

 icosahedron. If we add to these three figures the regular tetra- 

 hedron which we have just been studying, and the cube (which is 

 represented, at least in outline, in the skeleton of the hexactinelhd 

 sponges), we have completed the series of the five regular poly hedra 

 known to geometers, the Platonic bodies "f of the older mathema- 

 ticians. It is at first sight all the more remarkable that we should 

 here meet with the whole five regular polyhedra, when we remember 

 that, among the vast variety of crystalline forms known among 

 minerals, the regular dodecahedron and icosahedron, simple as they 

 are from the mathematical point of view never occur. Not only 

 do these latter never occur in crystallography but (as is explained 

 in textbooks of that science) it has been shewn that they cannot 

 occur, owing to the fact that their indices (or numbers expressing 

 the relation of the faces to the three primary axes) involve an 

 irrational quantity: whereas it is a fundamental law of crystallo- 

 graphy, involved in the whole theory of space-partitioning, that 

 "the indices of any and every face of a crystal are small whole 

 numbers J." At the same time, an imperfect pentagonal dodeca- 



* Of the many thousand figures in the hundred and forty plates of this beautifully 

 illustrated book, there is scarcely one which does not depict some subtle and 

 elegant yeornetrical configuration. 



t They were known long before Plato- llXdnov 5^ kuI iv tovtols irvdayopl^ei. 



X If the equation of any plane face of a crystal be written in the form 

 hx + ky -\-lz-\, then h, k, I are the indices of which we are speaking. They are 

 the reciprocals of the parameters, or reciprocals of the distances from the origin 

 at which the plane meets the several axes. In the case of the regular or pentagonal 

 do«lecahedron these indices are 2, 1 -r Vo, 0. Kepler described as follows, briefly 



