un certain Series ol' Sections 

 of the Regular Four-dimensional Hypersolids , 



BY 



ALICIA BOOLE STOTT, 



1. Ill making series of sections of the regular four-diuiciisional 

 figures by tlie method given in this paper it is only necessary to 

 knoAV tlie number of solids meeting at each vertex. The total number 

 in each figure can be found by counting the number of solids cut 

 in the sections. 



Taking the figures bounded by tetrahedra it is evident that a 

 section by a space cutting the edges meeting in a vertex at equal 

 distances from that vertex will give an equilateral triangular section 

 of each tetrahedron. Hence the complete section will be a three- 

 dimensional regular figure bounded by equilateral triangles. 



There are only three such figures, the tetrahedron, the octahedron 

 and the icosahedron ; so there will be no other four-dimensional 

 figure bounded by tetrahedra except those which have 4, 8 or 20 

 at each vertex. 



If groups of tetrahedra arranged so that there are 4, 8 or 20 

 round a point be cut by parallel spaces close enough together to 

 pass at least once through each edge, then the number of tetra- 

 hedra cut in the three groups respectively will be 5, 10 or bOO. 



Next taking cubes. The section of a cube by a space cutting the 

 edges meeting at a vertex at equal distances from that vertex is 

 an equilateral triangle. So there Avill be no other figure bounded 

 by cubes except those having 4, 8 or 20 at each vertex. But 8 



