IIEGULAE FOUR-DIMENSIONAL HYPEKSOLIDS. 5 



If the 8-cell were cut by a space JS parallel to ^ and passing 



thi-ough some point of tlie edge A K, eacli of the cubes N A, P A, 

 SA would be cut by a plane parallel to its surface of contact 

 with H A. The positions of these planes in the cubes could easily 



be determined after they have been brought into the space 8 . For 

 instance, if S' bisects A K, the sections of N A, P ^ and SA will 

 be squares parallel to C A, G A and E A through the midpoint of 

 A K. Similarly there will be square sections of the three cubes at H 

 and the complete section is a cube. Thus there will be three 

 cubes in this series: ]° the cube HA, 2° a cube bounded by 

 the sectional planes parallel to the faces of HA and 3° a cube 

 bounded by the squares P K, N K, S K 'à\\i\. the corresponding 

 faces of the cubes about H. This last cube is itself a solid of the 

 S-cell namely that opposite to HA ^). 



3. Definition 1. — Let a point at a distance n times AB from 

 A on the line AB be the point A,, B or B^_,^A. 



Definition 2. — Let pn be the projection, by a line parallel to 

 the base on to the perpendicular of an equilateral triangle, of a 

 length on the side equal to n times the side (jw = tV ]/^3). 



The 16 cell. 



4. Let ABCD, a tetrahedron in S , be one of the bounding 

 solids of a 1 G -cell. 



Li this figure there are S solids at each vertex, a condition 

 that will be satisfied if a tetrahedron be put on to each face of 

 ABCD, one to each edge and one to each vertex. Let the 

 vertices of those on ABC, BCD, CDA, DBA be D' , A', 

 B', C' respectively (fig. 2). Then those on the edges will be 

 A B C' D\ B CD'\4', CD A' B', D A B' C', A C B' D', B D A' C' . 



Four of the eight tetrahedra at A are represented in fig. 3. 



5. if the ] 0-cell be cut by a space S parallel to S 

 and passing through the point A^D, the tetrahedron ^ P C' _ö' will 



^) This way of dealing- with the 8-cell is given in the „Scientific Romances", No. 1, 

 „What is the fourth Dimension?" by C. H. Hinton, published by Swan, Sonnenschein & Co, 

 London, 1.S81, and the subsequent book of the same author: „The New Era of Thought" 

 London, 1888. Also in considering the 600-cell I received some valuable suggestions 

 from Mr. H. W. Cuijel. 



