3 G ANALYTICAL TR EATMENT OF THE POLTTOPES REGULARLY 



It is our aim to unearth in the following articles all the nets 

 of simplex extraction possible in space S n from n = 2 to n = 5 

 included. This task, concerned with new material, breaks up into 

 several parts. First we will have to deduce general characteristic 

 properties of the analytical symbols which are to represent the nets. 

 Secondly we will derive a simple rule solving the question under 

 what circumstances the symbols obtained do represents possible nets. 

 Thirdly application of this simple rule will lead to the knowledge 

 of all the possible nets and to a tabularization of them. Finally 

 we will pass in review the tabulated nets and devote some words 

 to an other method by which at least a part of these results can 

 be obtained. 



23. Theoretically speaking a net can be determined analytically 

 in two different ways, either as a whole or decomposed into its 

 constituent polytopes. So we will try to find either one symbol of 

 coordinates, representing all the vertices of the net at a time, or 

 in the case of pure nets one pair, in the case of mixed nets 

 several pairs of symbols, each pair consisting of a symbol repre- 

 senting all the vertices of any constituent and an other symbol 

 from which can be deduced all the centres of the repetitions of 

 that constituent in the same orientation occurring in the net. 



In order to blow life into this theoretical skeleton — forming 

 as it were a kind of working hypothesis — we consider the generally 

 known and simple case of the net of triangles in the plane. 



If we start (fig. 5) from a triangle A^ A 2 A 3 ==p â (i \ i. e. with 

 sides unity, and complete the three sides produced to three systems 

 of equidistant parallel lines, the distance of any two adjacent 

 parallel lines being the height of triangle p ( l\ we get the net N(p 3 ). 



From this generation it is at once evident that with respect to 

 the original p cl) 3 as triangle of coordinates all the vertices of the 

 net can be represented by the coordinate symbol (a i} a 2 , a 3 ), where 

 a i9 (i = 1, 2, 3) are any three integers for which Ha t =1. So 

 (a if a 2 , a ô ), lLa- t = 1 is the net symbol of N{p 3 ), under the condition 

 stated that a t are three integers. In this ever so simple case the 

 round brackets may be omitted, for the faculty of taking for 

 a { ,a 2 ,a 3 any set of three integers with a sum unity includes that 

 of interchanging the three digits. 



The net N(p$) consists of two sets of triangles, triangles p (i) 3 

 corresponding in orientation with A x A 2 A 3 and triangles p (i) 3 of 

 opposite orientation. If we consider only one of these two sets of 

 triangles and of these triangles only one of the three sets of homologous 



