24 ANALYTICAL TREATMENT OF THE POL1TOPES REGULARLY 



some new geometrical terms by the use of which the expression of 

 general laws will be simplified. 



In tig. 14 is represented in heavy lines one of the tC with 

 centre O and an eighth part of the M£ la) surrounding it, viz. that 

 part lying in the octant of the positive coordinates taken in the 

 directions OV\, OV 2 , OF B '. Now we make to correspond to the 

 different limiting elements of the surrounding cube the limiting 

 elements of the tC into which the first are transformed if the tC 

 is deduced from the surrounding cube by truncation at vertices, 

 edges and faces. So the triangle ABC of vertex import corresponds 

 to the vertex V, the edge A A' (or the face of edge import which 

 replaces it in an other case) corresponds to the edge VW X , the 

 octagonal face B'BCC' . . . corresponds to the face W 2 VW % . Then 

 by reflecting the triangle ABC into the three faces of M^ 2a) through 

 the corresponding vertex V as mirrors and by dealing in the same 

 way with the edge A A' with respect to the two faces through the 

 corresponding edge VW X and with the face B'BCC' . . . with respect 

 to the corresponding face W 2 VW B we get successively the eight 

 triangular faces of an BCD with V , the four upright edges of a P 4 

 with V it the two end planes of a P s with V\ as centre. We simplify 

 these expressions by saying that "multiplication" of the triangle ABC 

 round V, of the edge A A' round VW ± , of the face B'BCC' . . . round 

 /F 2 VW Z generates the indicated polyhedra RCO , P 4 = C, P 8 . 



In fig. 14 have been represented in ordinary lines the RCO 

 generated by the triangle ABC, the three cubes generated by the 

 edges AA' , BB' , CC' and the three P H generated by the faces 

 B'BCC' . ,C'CAA' . .,A'ABB' . . From this diagram it is clear 

 that the indicated RCO, C, P 8 fill up the interstitial space between 

 the tC, i. e. that the net bearing in Andreini's memoir the number 

 22 exists; we facilitate the inspection of this diagram by adding 

 a stereoscopic representation of it. x ) 



The deduction of the coordinate symbols of the new constituents 

 RCO, C, F 8 from those of the tC and its surrounding cube shows 

 us, what we have to do in general in order to obtain the coordinate 

 symbols of the new constituents. 



We begin with RCO obtained by multiplying the triangle ABC 

 round V. In order to get the representation of the triangle ABC 

 with respect to the original axes we have to replace the square 

 brackets of the symbol [1+^2,1+^.2,1] of tC by round 

 ones. In order to represent that triangle with respect to new axes 



*) The effect is enhanced if we place it so, as to have the small arrow at the left. 



