( 536 ) 



Mathematics. — " Foiirdimeiisional nets and their sections hy 

 spaces'. (First part). By Prof. P. H. Schoute. 



(Communicated in the meeting of January 25, 1908). 



Out of the table 



C^ .... 75° 31' 21", C,,.... 120^ C„„ .... 144° 

 63.... 90^ , C,, ....120°, Ce„„....164°28'39" 



of the angles formed bj two bounding bodies meeting in a face 

 of the regular cells of space Sp^ it is immediately evident that only 

 for the cells C^, C^g, C\^ can there be any question about each 

 respectively filling that space. It is well known, that this is really 

 the case. In the handbook included in the Sammlung Schubert 

 '' Mehr dimensionale Geometrie" (vol. II, page 241) is indicated how 

 the two nets of the cells C^^ and C,^ can be deduced by trans- 

 formation from the net of cells Cg, the existence of which is clear 

 in itself. We repeat this here in a somewhat different form to add 

 new considerations to it. 



1. The points with the coordinates (± J, =b 1, =t 1, ± 1) are the 

 vertices of an eightcell C^i with double the unit of length as length 

 of edge, the origin of the coordinates as centre and the directions 

 of the axes as directions of the edges. These vertices can be easily 

 arranged in two groups of eight points, one group of which contains 

 the points with a positive product of coordinates, the other group 

 the points with a negative one. Each of these groups has the property 

 that no two of the eight points are united -by an edge of Cf^\ 

 therefore we call them groups of non-adjacent vertices. Let us join 

 for each of these groups the two points lying in the same face of 

 C^-2) by a diagonal, then the systems of edges of two cells C^^^^' 

 are generated ; as each of the bounding cubes of C^p is circum- 

 scribed about one of the 16 bounding tetrahedra of each of the two 

 (7(21/2)^ we call these last inscribed in C^-'^, where one may be 



16 " 



called positive, the other negative. 



Let us now suppose the net of the Cg to be composed of alternate 

 white and black eightcells, so that two Cg with a common bounding 

 body differ in colour — from which it follows, that two C^ in 

 contact of edges do this too, whilst on the other hand two Cg in 

 face or in vertex contact bear the same colour — , and let us assume 

 that in each white Cg is inscribed a positive Ci, and in each black 

 Cg a negative one; then it is clear that both groups of C^ do not 



