8 ON CEETAIN SEEIES OF SECTIONS OF THE 



It also coincides -with the face {A B) {A F) {B F) of the octa- 

 hedron A {B F). The points {A B), (A F), {A E), {A C) are the 

 vertices of a square section of the octahedron A A'. This section 

 of the 24-cell, then, is bounded by f^ equilateral triangles, and 

 6 squares, the sides being eciual to the edges of the 24-cell. The 

 shape (fig. S) is that of the combination of octahedron and cube 

 in equilibrium. 



The octahedra already grouped about the octahedron A B give 

 4 solids at (AC), namely A A', A {CE), CC', A (BC); of these 

 the first two only are shewn in figure 6. 



Putting one on the face {A C) {CE) {A E) and one on the 

 face {A B) {AC) {B C) we have the required number 6 at {A C). 

 There will be similarly placed octahedra at {A B), {A F), etc. 



Another parallel space S" passing through the middle point of 



{A C) A' will give a square section of A A' and square sections of 

 the octahedra C C' and E E' . These three squares determine the 

 section of {A C) E'; it is a regular hexagon. This shape then 



is like the section by xS' . A space S parallel to S and passing 



through A' will coincide with the octahedron A B' C' D' E' F' . 



The 120-cell. 



8. The sections of this cell can be deduced in a similar man- 

 ner to that in which those of the 16-celland 24-cell were obtained. 



The plans are given in the diagrams VIII — XIV. 



The 600-ceII. 



3 



9. Let A B CD, a tetrahedron in /^ , be one of the bounding 

 solids of a 6()()-cell. In this figure there are 20 solids at each 

 vertex and this condition is satisfied at the vertices of A B C D \î 

 a tetrahedron be ])ut on to each of its faces, two to each edge 

 and 10 to each vertex. The bases of the tctrahedra in each of 

 these groups of 20 are the faces of an icosaliedron ^). Let FA 

 be that one which is bounded by the bases of the tetrahedra meet- 

 ing at A. Tlie vertex of the tetrahedron on ABC as base is F>' 

 and the vertex of the one on A C D' is {A C) ; the tetrahedron on 

 AD' {A C) is AB' {A C) {A J, likewise that on A {A C) {A J is 



A {A C) {A ) {A ) and that on A {A ) {A ) is ./ {A ) {A ) {A ). 



') The sections of such icosahedra appear as zones on the sections of the GOÜ-cell. 



