480 ON CONCRETIONS, SPICULES, ETC. [ch. 



with the whole five regular polyhedra, when we remember that, 

 among the vast variety of crystalUne 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 text-books 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*." At the same time, an imperfect pentagonal dodeca- 

 hedron, whose pentagonal sides are non-equilateral, is common 

 among crystals. If we may safely judge from Haeckel's figures, 

 the pentagonal dodecahedron of the Radiolarian is perfectly 

 regular, and we must presume, accordingly, that it is not brought 

 about by principles of space-partitioning similar to those which 

 manifest themselves in the phenomenon of crystalUsation. It 

 wall be observed that in all these radiolarian polyhedral shells, 

 the surface of each external facet is formed of a minute hexa- 

 gonal network, whose probable origin, in relation to a vesicular 

 structure, is such as we have alreadv discussed. 



In certain alhed Radiolaria (Fig. 232), which, hke the dodeca- 

 hedral form figured in Fig. 231, 5, have twenty radial spines, these 

 latter are commonly described as being arranged in a certain very 

 singular way. It is stated that their arrangement may be referred 



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

 hx + hy + lz= 1, 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 

 dodecahedron these indices are 2, 1 + ^5, 0. Kepler described as foUows, briefly 

 but adequately, the common characteristics of the dodecahedron and icosahedron : 

 "Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quin- 

 quanguUs figuratur expresse, hoc trianguUs quidem sed in quinquanguU forma m 

 coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli structura 

 perfici non potest sine proportione ilia, quam hodierni geometrae divinam appellant^' 

 {De nive sezangula (1611), Opera, ed. Frisch, vn, p. 723). Here Kepler was dealing, 

 somewhat after the manner of Sir Thomas Browne, with the mysteries of the 

 quincuns, and also of the hexagon; and was seeking for an explanation of the 

 mysterious or even mystical beauty of the 5-petalled or 3-petalled flower, — pulchri- 

 tudinis aut proprietatis figurae, quae animam harum plantarum character isavit. 



