DERIVED EROM THE REGULAR POLYTOPES. 67 



any two adjacent parallel spaces 8 n _ t being either the height of 8(n -\- 1) (1) 

 or twice that height, we get either the net n t or the net n u " 



"If we intersect a net of measure polytopes M n + i of space 8 n+i 

 by a space 8 n normal to a diagonal of a measure polytope and we 

 make that 8 n to pass either through a vertex or through the cen- 

 tre of an edge of that polytope we generate either a net tij or 

 a net n u . In order to obtain nets n l and n n with length of edge 

 unity we must start in the first case from a net N(M.$+i), in 



the second case from a net A(i/ n ^ 2 )." 



The first generation is easily proved, if we consider the cases of 

 the triangle net JV(p B ) and the triangle and hexagon net N(p z ,p 6 ) 

 of the plane and the cases of the net N(T, 0) and the net N{T, tT) 

 of threedimensional space first. 



But the second generation, used already in two different papers, 1 ) 

 has this great advantage that it furnishes at the same time an easy 

 method of deducing the character of the different constituents. We 

 only trace this method here, as the different constituents have been 

 found otherwise already. 



The generation itself shows that all the constituents are sections 

 of the measure polytope M n + i by a space 8 n normal to a diagonal. 

 In the first of the two papers quoted just now is demonstrated 

 that "the section of M n + i by a space 8 n normal to a diagonal 

 can always be regarded as a part of that space 8 n enclosed by 

 two definite, concentric, oppositely orientated, regular simplexes 

 8(n-\~l) of that space", i.e. that this section is a "regularly 

 truncated regular simplex". Moreover the second of the two papers 

 indicates how to find the amount of these truncations, whilst finally 

 the theorem V, or rather its inversion, teaches how to deduce the 

 zero symbol from the truncation numbers. 



F. Polarity. 



36. If we polarize one of the regular or one of the Archimedian 

 semiregular polyhedra with respect to any concentric sphere, i. e. 

 if we replace that polyhedron characterized by its vertices by the 

 polyhedron included by the polar planes of these vertices with respect 

 to that sphere, we pass from a body with one kind of vertex and 

 edges of the same length to a body with one kind of face and equal 

 dihedral angles. We suppose the simple laws of this "inversion" to 

 be known ; so we state only that the lines bearing the edges of the 



x ) Proceedings of the Academy of Amsterdam , vol. X , pp. 485 and 688. 



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