t 



DERIVED FROM THE REGULAR POLYTOPES. 9 



5. We finish the first part of this section by mentioning a theorem 

 already proved in the paper quoted in art. 1 , as this theorem will 

 be very useful in future. It is: 



"Any two spaces S k __ i9 $ n _ k containing together the vertices of 

 a regular simplex S {n -\~ 1) of S n are perfectly normal to each other." 



This theorem is an immediate consequence of the property that 

 any two edges without common end point determine a regular tetra- 

 hedron and are therefore at right angles to each other. For this 

 implies that, if S k _ i contains the vertices A i3 A 2 , . . .,A k and S n _ k 

 the vertices A k + ii A k + 2f . . . ,A n + i , each of the h — 1 indépendant 

 lines A i A t (/ = 2, 3, . . . , Jc) of S k _ i is normal to each of the n — k 

 independent lines A k bi A,, + tn (m = 2, 3, . . . , n — h-\-V) of S n _ u 1 ). 



B. The cltaracteristic numbers. 



6. We will now explain how the characteristic numbers of the 

 vertices, edges, faces, limiting bodies, etc. can be deduced from 

 the symbol of coordinates. 



The larger n is, the more elaborate the process becomes. So, in 

 order to divide the difficulties, we will begin by treating the cases 

 n = 4 and n = 5 at full length by means of an easy method, 

 working from two different sides, the vertex side and the limiting 

 polyhedron (n = 4) or limiting poly tope (n = 5) side, of the series 

 vertex, edge, face, etc. Afterwards we will show a more direct way 

 leading immediately to the knowledge of the forms and the numbers 

 of the different kinds of limits (l) p of p dimensions. 



In the cas3s n = 4 and n = 5 of the four and the five charac- 

 teristic numbers we determine for itself the first two and the last 

 two, using the law of Euler for n = 4 as a check and for n = 5 

 as a means of finding the lacking middle number of the faces. 



The number of vertices is easily found in all possible cases. If 

 all the n -f- 1 digits of the symbol of coordinates of the pol}/ tope 

 in 8 n are different it is (n -\- 1)! This number {n — | — 1)1 must be 

 divided by 2! for any two, by 3! for any three, by 4! for any 

 four digits being equal, etc. 



The number of edges can be calculated as soon as we know how 

 many edges pass through each vertex. For the product of this latter 

 number by the number of the vertices indicates how many times 



J ) Compare the theorem of art. 30 (on p. 42 of the first volume of my textbook, 

 "Mehrdimensionale Geometrie") where the "two groups of lines through Ö" may be 

 replaced by "one group of lines through (here A x ) and an other group of lines 

 through 0' (here A k+1 )" 



