43 . 



number of them, it is evident that all these points must be 

 distributed over the whole surface of the sphere in such a way that 

 all these polygons are arranged as the faces of a regular polyhedron 

 inscribed in the sphere, - - the regular polyhedron formed by 

 all the points A being the polar figure of the regular polyhedron 

 formed by all points B as corners, and conversely. Now it is well- 

 known, that there are only five possible regular, endospherical poly- 

 hedra 1 ) : the tetrahedron, the cube, the octahedron, the dodecahedron, 

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

 polyhedra. Indeed, these 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 

 jig. 42, A is a ternary axis ; the three axes B may be quinary ones ; etc. 



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 cor- 

 responding 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 men- 

 tioned, 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 purpose 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 another 



2?r 2;r 

 general property of systems with several axes of the periods and . 



2fl- n p 



Let ON (fig. 43) be an axis of the period , and OP another with 

 n 



l ) As to the case of = oo , we may refer to our previous remark (p. 35). 



