10 ON CEKÏAIN SERIES OF SECTIONS OF THE 



This projection is shewn in fig. IS, where R, M, 0, P are the 

 points C', {A_X {A C), C, and N is tlie middle point of (J ) {A ), 



H V is drawn parallel to JIN and V \^ the distance from A C 

 at Avhich the plane cnts C{AC). Let VO be a times P and we 

 have the points {^l B) B, {A C) C, {A IJ) 7); for the remaining ver- 

 tices of this form see list of vertices and diagram T. 



Again S and I D' intersect in a plane passing through the 

 points {C A) C, {CA) A, {A B) A, {A B) B, {B C) B, {B C) C, of 



'- II a a II II II 



fig. 12. 



Let BMOF, fig. is, be a projection of IB' on a plane 

 through {AB){D',;) [C ^) C. Then Bq = pa and qV^ is parallel 



to M N. It will be found that V P = a times B . whence 



1 



Ave have the points C (C ), A (A ), B {B ) of fig. 12. 



'- II ^ D^ II ^ U II ^ d' ^ 



From this it will be seen that the tetrahedron ,/ {A ) [A ) {A \ 

 is cut in the points ^ {A ), A {A ), A (A ) (fig. 13, ^j ^ /•); li^^ewise 

 A {AC) {A) {A ) is cut in the points A {A ), A {A ), {AC) A 

 (fig. 13, pre-) and A B' {A C) {A ) is cut in the points {A C) A, 

 A {A),D' (fig. 13, srB'). 



a Ü 



The tetrahedron A C B' {A C) is cut in the points {A C) A 



{A C) C, B' (fig. 13, 8t B'). 



II '~ 



Here we have a section of a tetrahedron of each type given in 



Fig. 10 except ABCB which is not cut by /tS^ , and as the tetra- 



hedra of each type are all similarly related to A B C B their sec- 

 tions will be e(pial. 



In constructing the model of this section of the GOO-cell there 

 will be four equilateral triangles (/> q r) related to each other as 

 the vertices of a tetrahedron. 



The sides of the equilateral triangles are the bases of isosceles 

 triangles {p r s, fig. 13) and the sides of these isosceles triangles 

 are the bases of other isosceles triangles (*• r B'). The last isosceles 

 triangles are arranged in pairs with a common side, and the other 

 sides are the sides of isosceles triangles {s t B'). 



One of the four similar regions of this model is given in diagram 

 1 {N,M',L')\ these regions are connected by triangles {M). 



3 



11. Let the 600-cell be cut by a space /S' passing through 

 {A B), {A C), {A B), {B C), {B B), {C B). This is parallel to s'^. Here 



