REGULAR FOUR-DLMENSONAL HYPERSOLIDS. 9 



Thus (A) {A^) {A ) is the face opposite to BCD on lA 



(fig. 9 and 10). etc. 



The five tctraliedra just given are all difFerently related to A B C D 

 and, if thev be taken as types, the vertices of the remaining tetra- 

 hedra about A B CD may at once be written down and can then 

 easily be placed in space. For instance there are four of the form 

 ABCD'. The form A C B' {A C) gives two tetrahedra on AC 

 namely A C B' {A C) and A CD' [A C) and also two on each of 

 the other edges as B C Di {B C) and B C Â {B C) on B C, and 

 so on. The form ./ D' {A C) {A ) gives six tetrahedra touching A 



arranged in pairs: 



AD' {AC) (A^) and AD' {AB){A^), 

 A C' {AB) {A J „ AC' {AD) {A ^), 

 AB' {AC) {A^) „ A B' {AD) {A J 



and also those about B, C, and D. So A {A C) {A ) {A _) gives 

 three tetrahedra about ./ namely A {A C) {A ^ {A p, A {AB) {A .) (./^) 

 and A {AD) {A ^) [A ^y, about B are B {B C) {B ) {B ) and so on. 

 There are also four of the form J(./ ) (.7 ) (A ). The remainiuff 

 vertices of the GOü-cell are named as follows: A is the vertex of 



the tetrahedron on {A ) {A ) {A ), {A'B) is the vertex of the tetra- 

 hedron on {A B) {A ) {A ) and A' {A' ) {A' ) {A' ) is related to 



ABCD as A {A ) {A ) {d ) is to ABCD (see list of vertices). 



/; c ij ^ ' 



The tetrahedron on the side of the GOO-cell opposite to AB CD 

 is ÛJ jS 7 (J , the lines drawn from A to u., from B to (2 and so on 

 l)eing diameters of the figure. 



'I'he arrangement of teti'ahedra about a. ^ y è is similar to that 

 about ABCD, so tliat the vertices o})posite to those already given 

 may be written down by simply changing A into u, B into /3, 6' 

 into y, D into è. 



10. Let the 600-cell be cut by a space S passing through the 

 points Â B' CD'. It will be parallel to s\ Now .v'^ and I A 



intersect in a plane passing through the points B' , C', D', fig. 11. 

 To find where this plane cuts the lines ^ (J 7^), C{A C), D {A D), 

 let 1 A be projected on a plane passing through C', {A ), {A C), C. 



