IX] OF JOHANNES MULLER'S LAW 729 



icosahedron, are conjugate pairs; but the first and simplest of all 

 solid figures, the tetrahedron, has no conjugate but itself. 



In our httle shell of Dorataspis, the twenty spicules (as we have 

 seen) spring from and correspond to the twenty facets of the poly- 

 hedron, twelve pentagonal and eight hexagonal, meeting at thirty-six 

 corners, in all cases three by three; we may write the formula of 

 the polyhedron, accordingly, as 



12i^5 + 8i?'6 + 36C3. 



If we now connect up the twenty spicules, three by three, we 

 shall obtain thirty-six triangles, completely covering the figure; 

 but we shall find that of the twenty corners twelve are surrounded 

 by five, and eight by six triangles. The formula is now 



362^3 +I2C5+8C6, 



and the two figures are fully reciprocal or conjugate. 



I do not know of any radiolarian in which this configuration is 

 to be found; nor does it seem a likely one, owing to the large and 

 variable number of edges which meet in its corners. But we may 

 have polyhedra related to, or derived from, one another in a less 

 full and perfect degree. For instance, letting the twenty spicules 

 of Dorataspis again serve as corners for the new figure, let four 

 facets meet in each corner; or (which comes to the same thing) let 

 each spicule give off four branches or offshoots, which shall meet 

 their corresponding neighbours, and form the boundary-edges of a 

 new network of facets. The result (Fig. 34'3) is a symmetrical 

 figure, not geometrically perfect but elegant in its own way, which 

 we recognise in a number of described forms*. It shews eight 

 triangular and fourteen rhomboidal facets; and its formula is 



8^3+142^4 + 2004. 



Many subsidiary varieties may arise in turn: when, for instance, 

 certain of the little branches fail to meet, or others grow large and 

 widely confluent, always in symmetrical fashion. 



We now see how in all such cases as these there is a double 

 symmetry involved, that of two superimposed, and conjugate or 

 semi-conjugate, figures. And the ambiguity which attends such 

 descriptions as that which Johannes Miiller embodied in his "law" 



* Cf. W. Mielck, Acanthometren aus Neu-Pommern, Diss., Kiel, 1907. 



