EEGULAE FOUE DIMENSIONAL HYPEESOLIDS. 7 



put on to each face and one to each vertex oî A B C D JtJ F. By 

 an inspection of tigui'e 6 it is clear that 



oil the face J JB C k the octahedron AB C {A B) {B C) {C A), 



„ „ „ ACE,, „ „ ACE {AC) {CE) {E A), 



„ „ „ AEF„ „ „ AEF (AE) (EF) (FA), 



„ „ „ AFB „ „ „ AFB (AF) (FB) (BA), 



„ „ „ BBC „ „ „ BBC {DB) {B C) (CB), 



„ „ „ BCE„ „ „ BCEiBC) (CE) (ED), 



„ „ „ BEF „ „ „ BEF{BE) {EF)(FB), 



„ „ „ BFB „ „ „ BFB{BF) {FB){BB). 



In the same manner tlie octahedron 



on A k A {A B) {A C) {A E) {A F) A', 



„ B „ B (AB) (BC) [BB) {BF) B', 



„ C „ C {AC) {BC) {CD) {CE) C' , 



„ B „ B{BB) {CD) {BE) {B F) B' , 



„ E „ E {A E) {CE) {BE) {E F) E', 



„ F „ F {A E) {BF) {BF) {E F) F'. 



7. If the 24-cell be cut by a space S parallel to -5' and pas- 

 sing through the point of bisection of A {A C) the octahedron yl {CE) 

 will be cut in a plane parallel to A CE, and passing through the 

 points of bisection of C {A C), C{CE), E {A E), A {A E), E {CE); 

 the section will be a regular hexagon {a be de/). 



Similarly the sections by ^ of the remaining seven octahedra 

 on the faces of ABC BEF are reguhir hexagons. 



Now in the octahedron A {CE) we see that S passes through 

 the ])oints A^ {A C), A^ {A E) and in the octahedron A {F B) we 



see that it passes through A^ {A B) and A^ {A F). But the lines 



A {A C), A {A E) and A {A ~F), A {A B) are also edges of the octa- 

 hedron A A', whence we find that the section by /S of this octa- 

 hedron is a square with side equal to half the edge of the 24-cell. 

 There will be similar square sections of the octahedra on B, C, B, E, F. 

 The shape is shewn in fig. 7 ; it is a combination of octahedron 

 and cube, in the crystallographic sense, the octahedron predominating. 



3 3 



Let S^^ be a space parallel to S and passing through {A C), it 



will also ])ass through {CE), {A E). So that it coincides with that 

 face of the octahedron A {CE). 



