28 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



No overlapping and no hole. By a translation al motion in the 

 direction of one of the axes over a distance 2a the system of 

 vertices T) is transformed in itself; so, if the central measure poly- 



tope [a, a, . . . , a] is filled exactly by the set of constituents found 

 above, these constituents form a net By a reflection in one of the 

 n spaces œ { = 0, (i= 1, 2, . . . 9 n), the system T) also is transformed 

 in itself; so, if the part of the central measure polytope M n {2n) 

 containing the points with positive coordinates only is filled exactly, 

 the constituents form a net. We indicate this part of the central 

 measure polytope by the symbol 7¥ n (+fl) . 



We now prove the following lemma: 



„Let (P) H a be a constituent lying partially within M^ +a) and 

 (P)n'_î any of its limits lying partially within M^ +a) . Then the 

 set of polytopes obtained above always contains one and only one 

 polytope (P),f having with (P),/' the limit (P)n'-\ in common; this 

 (P) H /; lies with respect to (P)„ on the opposite side of {P)*^" 



The condition that (P)„ a lies at least partially within J/ n (+a) is 

 fulfilled, if we consider that repetition of the chosen constituent 

 the coordinates of the centre of which admit the values -|~ a and 

 zero only. We find, if all the coordinates are zero the groundform 

 contained in T), if all the coordinates are -f- a a polytope contained 

 in T), if some coordinates are -\- a and the other ones zero a 

 polytope contained in T''). Now the first case, of the groundform, 

 and the second case, of all coordinates = -\- a, are included in 

 the third case, as we get them by putting k = and k = n. 

 So we can choose for (P) /( a the polytope 



[a-\-a— a ki a + a—a^i,. . . , a-\~a—ai][a. k+u a k+2 , . . . , a n _ i9 A n ] 



W± y c#2j • • • • • • j ^k ^k+if <&k+2i ?^n 



where the x t placed under the two syllables indicate the coordinates 

 to which the two sets of digits refer, and occupy ourselves with 

 the question how to get a limit (/) n _ 4 of this prismotope. Now in 

 general the limits {l) n _\ of the prismotope (P,.; P n _ fe ) present them- 

 selves in two groups, viz. if (P)/,_ ( is any limit (/)/,._-, of (P),. and 

 (P) n -k-i any limit (/) n _ fc _ d of (P) H _ A ., in the two forms (P fc _ i; P n _ k ) 

 and (P, c ; P n _ k _ ± ). So 1 ) 9 we have to consider the two different cases 



') For a limit (P) ' lying at least partially within M none of the coordinates 



may assume values equal to or surpassing + a for a ^ the vertices of that limit; therefore 

 in the first case (P/ C _i ; P n —/ C ) we have to place between round brackets a certain 

 number s y of the largest digits \a + « — «<1 where a — a \ is taken with the reversed 

 sign, i. e. a k a / ._ 1 ,..., a fc _ 8 , ^ taken in inverted order. 



