IX] OF JOHANNES MtJLLER'S LAW 725 



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 (Circorhegma) is 

 perfectly regular, and we may rest assured, accordingly, that it is 

 not brought about by principles of space-partitioning similar to 

 those which manifest themselves in the phenomenon of crystallisa- 

 tion. It will be observed that in all these radiolarian polyhedral 

 shells, the surface of each external facet is formed of a minute 

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

 structure, is such as we have already discussed. 



In certain allied Radiolaria of the family Acanthometridae (Fig. 

 341), which have twenty radial spines, the arrangement of these 

 spines is commonly described in a somewhat singular way. The 

 twenty spines are referred to five whorls of four spines each, arranged 

 as parallel circles on the sphere, and corresponding to the equator, 

 the tropics and the polar circles. This rule was laid down by the 

 celebrated Johannes Miiller, and has ever since been used and 

 quoted as Miiller's law*. But when we come to examine the figure, 

 we find that Miiller's law hardly does justice to the facts, and seems 

 to overlook a simpler symmetry. We see in the first place that 

 here, unlike our former cases, the twenty radial spines issue through 

 the facets (and all the facets) of the polyhedron, instead of coming 

 from its corners; and that our twenty .spines correspond, therefore, 

 not to the corners of a dodecahedron, but to the facets of some sort 

 of an icosahedron. We see, in the next place, that this icosahedron 

 is composed of faces of two kinds, hexagonal and pentagonal ; and 

 that the whole figure may be described as a hexagonal prism, whose 

 twelve corners are truncated, and replaced by pentagonal facets. 

 Both hexagons and pentagons appear to be equilateral, but if we 



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

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

 quangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam 

 coaptati^. Utriusque horum corporum Jpsiusque adeo quinquanguli structura 

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

 (De nive sexangula (1611), Opera, ed. Fritsch, vn, p. 723). Here Kepler was dealing, 

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

 quincunx, 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 fgurae, quae animam harum plantarum characterisavit. 



* See Johannes Miiller, Ueber die Thalassicollen, Polycistinen und Acantho- 

 metren des Mittelmeeres, Abh. d. Akad. Wiss. Berlin, 1858, pp. 1-62, 11 pi. 



