8 ANALYTICAL TREATMENT OF THE POLYTOPES EEGULARLY 



In general a form obtained in this way may present itself in two 

 different positions with respect to the simplex S {n 4~ 1) of coor- 

 dinates. So we may write e. g. for (21100) also (0 — 1 — 1 — 2 — 2), 

 or, if we invert the sign of all the coordinates and indicate that 

 we have done so by putting the sign minus before the brackets, 

 — (01122), i.e. —(22110); so (21100) = — (22110) and likewise 

 (22110) = — (21100). Really the symbols with the values satisfying 

 the condition 2^=1 corresponding to (21100) and (01122) are 

 (iff — Î — f^ aR d ( — 10011) representing, if we omit the 

 brackets, two points P, P situated symmetrically to each other with 

 respect to the centre of gravity G5 -5- -5 5- -5) of the simplex S (5) of 

 coordinates. So (22110) is the form (21100) in opposite orientation; 

 in the equation (22110) = — (21100) the difference in orientation is 

 indicated by the sign minus. 



The forms the symbols of which are not affected by the "inversion" 

 mentioned are marked by an asterisk ; as they do not alter as a whole when 

 they are put into the opposite position they possess central symmetry x ). 



4. The results obtained show that the geometrical method followed 

 by M rs . Stott and the analytical method developed here cover 

 exactly the same ground, i. e. that they lead up to the same system 

 of forms. Nevertheless we should jump to a wrong conclusion, if we 

 deduced from this coincidence of results that by either of the methods 

 all the possible forms with one kind of vertex and one length of edge 

 have been found. We show this by remarking that the combina- 

 tion of the two zero symbols (1000) and (1110), or of the proper 

 values (1000) and (-| 1 -i- — -1), of T and — T gives us the vertices 

 of the cube, which implies that all the forms deduced from the cube 

 by M rs . Stott can be represented by couples of symbols in bary- 

 centric coordinates as derived from the tetrahedron 2 ). 



x ) For the deduction of the e and c symbols from the symbols of coordinates and 

 reversely compare the part D of this section. 



The first of the tables added at the end of this memoir is destined to put on record 

 for n = 3, 4, 5 the different polyhedra and polytopes deduced from the simplex with 

 their principal properties. Of this table the first column contains M^. Stott's symbol, 

 the second the symbol of coordinates and the third the value by which the coordinates 

 have to be diminished in order to find the true coordinate values for which S xi = 1. 

 The following columns will be explained farther on. 



2 ) So the most complicated form e 1 e 2 C=tCO can be represented by 

 f 7 + 3' 3 — 1' — • 1 — 1' —5 — 1" 

 V 4 ' 4 ' 4 ' 4 



f 7-3' 3 + 1' — 1 -hl' —5 + l^ i 1 

 \ 4 » 4 ' 4 "' 4 J) 



if 1' and 3' stand for 1/2 and 3 1/2. 



