at first sight all the more remarkable that we should here meet {480} 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 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 crystallography, involved in the whole theory of space-partitioning, that “the indices of any and every face of a crystal are small whole numbers[490].” At the same time, an imperfect pentagonal dodecahedron, 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 crystallisation. 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 (Fig. [232]), which, like the dodecahedral 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 {481} to a series of five parallel circles on the sphere, corresponding to the equator (c), the tropics (b, d) and the polar circles (a, e); and that beginning with four equidistant spines in the equator, we have alternating whorls of four, radiating outwards from the sphere in each of the other parallel zones. This rule was laid down by the celebrated Johannes Müller, and has ever since been used and quoted as Müller’s law. The chief point in this alleged arrangement which strikes us at first sight as very curious, is that there is said to be no spine at either pole; and when we come to examine carefully the figure of the organism, we find that the received
Fig. 232. Dorataspis sp.; diagrammatic.
description does not do justice to the facts. We see, in the first place, from such figures as Figs. [232], 234, that here, unlike our former cases, the radial spines issue through the facets (and through all the facets) of the polyhedron, instead of through its solid angles; and accordingly, that our twenty spines correspond (not, as before, to a dodecahedron) but to some sort of an icosahedron. We see in the next place, that this icosahedron is composed of faces, or plates, of two different kinds, some hexagonal and some pentagonal; and when we look closer, we discover that the whole figure is that of a hexagonal prism, whose twelve solid angles are replaced by pentagonal facets. Both hexagons and pentagons {482} appear to be perfectly equilateral, but if we try to construct a plane-sided polyhedron of this kind, we soon find that it is impossible; for into the angles between the six equatorial hexagons those of the six united pentagons will not fit. The figure however can be easily constructed if we replace the straight edges (or some of them) by curves, and the plane facets by corresponding, slightly curved, surfaces. The true symmetry of this figure, then, is hexagonal, with a polar axis, produced into two polar spicules; with six equatorial spicules, or rays; and with two sets of six spicular rays, interposed between the polar axis and the equatorial rays, and alternating in position with the latter.
Müller’s description was emended by Brandt, and what is now known as “Brandt’s law,” viz. that the symmetry consists of two polar rays, and three whorls of six each, coincides with the above description so far as the spicular axes go: save only that Brandt specifically states that the intermediate whorls stand equidistant between the equator and the poles, i.e. in latitude 45°. While not far from the truth, this statement is not exact; for according to the geometry of the figure, the intermediate cycles obviously stand in a slightly higher latitude, but this latitude I have not attempted to determine; for the calculation seems to be a little troublesome owing to the curvature of the sides of the figure, and the enquiring mathematician will perform it more easily than I. Brandt, if I understand him rightly, did not propose his “law” as a substitute for Müller’s law, but as a second law applicable to a few particular cases. I on the other hand can find no case to which Müller’s law properly applies.
If we construct such a polyhedron, and set it in the position of Fig. [232], B, we shall easily see that it is capable of explanation (though improperly) in accordance with Müller’s law; for the four equatorial rays of Müller (c) now correspond to the two polar and to two opposite equatorial facets of our polyhedron: the four “polar” rays of Müller (a or e) correspond to two adjacent hexagons and two intermediate pentagons of the figure: and Müller’s “tropical” rays (b or d) are those which emanate from the remaining four pentagonal facets, in each half of the figure. In some cases, such as Haeckel’s Phatnaspis cristata (Fig. [233]), we have an ellipsoidal body, from which the spines emerge in the order described, but which is not obviously divided by facets. In Fig. [234] I have indicated the facets corresponding to the rays, and dividing the surface in the usual symmetrical way. {483}
Fig. 233. Phatnaspis cristata, Hkl.
