482 ON CONCRETIONS, SPICULES, ETC. [ch. ix 



10 



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, sHghtly 

 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. 



Miiller'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 shghtly 

 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 Miiller's law, but as a second law applicable to a few 

 particular cases. I on the other hand can find no case to which Miiller'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 Miiller's law; for the 

 four equatorial rays of Miiller (c) now correspond to the two polar 

 and to two opposite equatorial facets of our polyhedron : the 

 four "polar" rays of Miiller (a or e) correspond to two adjacent 

 hexagons and two intermediate pentagons of the figure: and 

 Miiller's "tropical" rays (6 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 elUpsoidal 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. 



