60 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



What is said above about contact of constituents of different 

 groups by a limit (/)„_i is a mere application 1 ) of theorem XVI. 



As the net of measure polytopes M n of S a shows, the //-dimen- 

 sional space round any point contains 2" right angles, if the 

 //-dimensional angle of M n is called a right one. As all the //-dimen- 

 sional angles of the self space filler are equal and // -(- 1 of these 

 polytopes concur in a vertex of the net, the //-dimensional angle of 



2" 

 the self space tiller is — r — right angles. 



n-\- I 



Theorem XVIII. "The net of S n with the period unity admits 



p n — p + 1 



the // constituents (II . . 1 00 . . 0), (p = 1, 2, . . . ,n) consisting 

 of — constituents in both orientations for // even and of 



2 2 



constituents in both orientations and one central symmetric consti- 

 tuent for // odd". 



Theorem XIX. "The net of JS n with the partition cycle 1, // 



n — p 



admits the n- 1 constituents (1 00 . . 0), (22 . . 2 1 00 . . 0) for 



// 



p = l 9 2, . .,ii — 1 and (II . . 10) consisting of — constituents in 



both orientations and one central symmetric constituent for // even 



// 4- 1 

 and of -—- constituents in both orientations for n odd". 



2 



These theorems immediately follow by specializing the general 

 results. We give them here expressis verbis as we will indicate later 

 on an other deduction of them. 2 ) 



32. A survey of the results for // = 2, 3, . . . , 7 suggests one or 

 two general remarks. 



The first can be stated in the form of: 



Theorem XX. "Every simplex polytope partakes in the formation 

 of two nets. This is true without any reserve for the central sym- 

 metric constituents, it is also true for each of the two different 

 positions of an asymmetric constituent." 



') It is an easy task to demonstrate theorem XVII by itself by showing that the 

 image points of the centre of the central polytope with respect to the spaces S n _ i 

 bearing the limits (On— 1 as mirrors form the centres of the polytopes in n — 1-dimen- 

 sional contact with the central polytope. 'We consider this verification as a useful exer- 

 cise, even in the special case n = 3 of ordinary space. 



2 ) Though we do not wish to push the general investigation any further we still 

 mention the following theorem : 



"The net of .9 2 » 1 with the power partition cycle 2 W is built up of two central sym- 

 metic constituents only". 



