42 



hedra *) : the tetrahedron, the cube, the octahedron, the dodecahedron, 

 and the icosahedron, these being the so-called "platonic" regular 

 polyhedra. Ind, eedthese polyhedra represent together three pairs 

 of polar figures ; for the cube and the octahedron, and also the pen- 

 tagonal dodecahedron and the icosahedron, are pairwise polar forms 

 of each other, while the tetrahedron has itself as polar figure. In 

 fig. 40, A is a ternary axis; the three axes B are quinary ones. 

 Thus it follows from this, that because every pair of polyhedra 

 corresponds to the same system of axes (A and B being simply 

 interchanged), there are only three new symmetry-groups of this 

 kind possible, namely- those corresponding to the directions of 

 the straight lines which in the .tetrahedron, in the cube, and in 

 the pentagonal dodecahedron join the geometrical centre of each 

 with its corners, and with the centres of the limiting polygons. 

 We shall call the groups mentioned, in consequence of their relations 

 to the endospherical polyhedra, the tetrahedron-group T, the cube- 

 group K, and the pentagonal dodecahedron-group P. Of course the 

 octahedron and the icosahedron might be chosen for this purpos 

 as well; the choice made is quite arbitrary and of no interest, 

 because the result is always the same. 

 7. Before reviewing the corresponding symmetry-elements of 



these three groups, it appears 

 advisable first to consider ano- 

 ther general property of systems 

 with several axes of the periods 



27T . 27T 



- and -;-. 

 n p 



Let ON (fig. 41) be an axis of 



27T 



the period , and OP another 

 n 



P' 27T 



with the period . By rotating 



the figure round ON through its 

 Fig. 41. characteristic angle, ON remains 



unchanged in space, but OP 



coincides with an equivalent axis OP'. If now the figure is turned 

 round OP, ON will coincide with a similar axis ON'. The successive 

 rotations round ON and OP have therefore the final effect that 



As to the case of n = oo , we may refer to our previous remark (p. 34) . 



