( 67 ) 



the four positions 12,13,14,15 of the origiiial vertex 1 are the 

 vertices of a tetrahedron of opposite orientation, the rotation of wiiich 

 provides us with the 5 7' of vertex import. The relation between these 

 two sets of 57' can be indicated by saying-, that two T of the same 

 set have nothing at all and that two T of different sets can have 

 only one vertex in common. 



If we had followed here the notation indicated under e^ and e^ 

 each vertex would have had to hear four digits, the digit of the 

 original vertex of ;S(5) followed by the digits of the other vertices 

 of the 7' witii which the vertex is moved out ; however, for short- 

 ness we have placed after the digit of the vertex the only digit 

 which does not occur at the vertices of the 7' moved out. 



In this new notation of pairs of digits, where — at variance with 

 the notation applied under c\ — the order of succession is of 

 influence, the ten P^ of edge import present themselves in two 

 quintuples, which can be obtained by putting after each of the 

 digits of the pair of digits of an edge successively each of the three 

 remaining digits; so 43 gives the three edges (41,31), (42, 32), 

 (45,35) of the P^ of fig. 8 turned upside down, while 52 leads in the 

 same way to (51, 21), (53, 23), (54, 24), the parallel edges of the P, 

 of fig. 9 turned upside down. Similarly the ten P^ of face import 

 are found by putting after each of the three digits of a face of 7' 

 successively one of the two remaining digits ; so 125 gives the two 

 endplanes (13, 23, 53), (14, 24, 54) of the P, of fig. 8, 134 the two 

 endplanes (12, 32, 42), (15, 35, 45) of the P, of fig. 9. 



The limiting polygon is a regular decagon with side s: from this 

 ensues the possibility of drawing the ten 7' immediately in position. 

 The 20 vertices are arranged in two wreaths (10, 10), of regular 

 decagons. 



e,e, aS(5) — (60, 120, 80, 20) — {ot 0, —, 10P„ 5tT). 



In this case, for which fig. 11 represents the result, the 7' of 

 fig. 2 is transformed into a tO (fig. 12); of the triplet of digits placed 

 at each ^'ertex of this tO the first indicates the original vertex of 2\ 

 the second with the first the edge moved out, the third with the 

 two preceding ones the face moved out. This notation with triplets 

 of digits differs again from that applied in fig. 5 in this that the 

 order of succession of the second and third digits, of no consequence 

 there, is of infiuence here. 



If we have traced in fig. 11 the t(J of fig. 12, rotation about the 

 centre, accompanied by an addition of unity to all the digits, gives 



5* 



