DERIVED FROM THE REGULAR POLYTOPES. 



59 



I 



8 



1 



6 





II 



17 





8 





III 



26 



2 



6 





IV 



35 





8 





V 



44 



2 



2 





VI 



116 



2 



6 





VII 



/III 



125 

 134 







8 

 8 



IX 



224 



2 









X 



233 



2 



Ü 





XI 



XII 



XIII 



XIV 



XV 



XVI 



XVII 



XVIII 



XIX 



XX 



n 



1115 

 1124 

 1214 

 1133 

 1313 

 1223 

 1232 

 2222 

 11114 

 11123 



= 7 



2 



8 



S 



XXI 



XXII 



XXIII 



XXIV 



XXV 



XXVI 



XXVII 



XXVIII 



XXIX 



11213 



11222 



12122 



111113 



1 11122 



111212 



112112 



1111112 



11111111 



2 



8 



Under n = 6 no cases of a power partition cycle (except 6 vy/i , 

 the self space filler) present themselves, as n -\- 1 is prime here. 

 For 72 = 7 we find besides 1 X xix still 7 y , 7 xy , 1 XXVII wiih v = % 

 and 1 xviii W1 ^h v = 4>. 



Instead of pushing this general investigation any further we will 

 give here the generalizations of the three nets of the plane to 

 space S n . 



Theorem XVII. "In space S n the central symmetric polytope 

 with the zero symbol (n, n — 1, n — 2, ..., 1, 0), represented 

 also by the expansion symbol e i e 2 e 3 ... e n _ 2 e n-i &( n ~\~ 1) > is the 

 only self space filler of simplex extraction. This unique geometric 

 constituent of the net presents itself in n -j- 1 different groups with 

 the property that the vertices of the constituents of each group 

 form the vertices of the net, each vertex taken once, in other 

 words: that no two constituents of the same group have a vertex 

 in common. In this "cycle of constituents" (compare the footnote 

 of art. 29) formed by these groups G , G u . . ., G n any polytope 

 of the group G k is touched along its limits y of vertex import 

 by (^ -j- 1 )i polytopes of group G k _ 1} along its limits g x of edge 

 import by {n-\~l) 2 polytopes of group G k _ 2 , etc. So, in order 

 to perform the task of colouring the polytopes of this net in such 

 a way that any two polytopes bearing the same colour are free 

 from each other (a polydimensional bud of the renowned shrub "map 

 colouring") it will be necessary and sufficient to have at hand 

 n-\-l different paints, one for the polytopes of each group". 



an 



'The ^-dimensional angle of the self space filler of S n is 

 right ones". 



n 



+i 



