Analytical treatment of the polytopes regularly derived 



from the regular polytopes. 



BY 



P. H. SCHOUTE. 



INTRODUCTION. 



In a memoir recently published by this Academy (Verhandelingen, 

 vol. XI, n°. 1) M rs . A. Boole Stott has given the geometrical treat- 

 ment of the polytopes regularly derived from the regular polytopes 

 of poly dimensional space. In these pages I wish to complete her 

 beautiful considerations by giving the analytical counterpart x ). 



The basis of this analytical counterpart is the fact that the coor- 

 dinates of the vertices of the tetrahedron may be represented by one 

 of the symbols (1, 0, 0, 0) and -J-[l, 1, 1], while those of the ver- 

 tices of cube, octahedron and icosahedron can be put in the forms 

 [1, 1, 1], [1, 0, 0] and [1 -f- ]/~5, 2, 0] : 2 respectively. The meaning 

 of these symbols will be explained later on. 



This paper is divided into five sections. In the first, concerned 

 with the offspring of the regular simplex, we will meet chiefly the 

 amplifications of the symbol (1, 0, 0, 0) of the tetrahedron. The 

 second and the third, dealing in the same manner with the measure 

 polytope and the cross poly tope, will bring us chiefly amplifications 

 of the symbols [1, 1, 1] and [1, 0, 0] of cube and octahedron. The 

 fourth will deal with the half measure polytopes and allied forms 

 represented by amplifications of the symbol \\\ , 1, 1]. Finally in 

 the fifth section about the extra regular polyhedra and polytopes 

 we will have to use the symbol of the icosahedron. 



*) I had the great advantage of reading the original manuscript to Mis. Stott; the 

 ensuing discussion. — I acknowledge this with thankfulness — has led to a simplification 

 of the proofs of several of the theorems. 



1* 



