14 ON CERTAIN SEIUKS OF SECTIONS OF THE 



,:i 



1(). Let the ()()() cell be cut by a space /S'^ through (///3), (/i 7), 



(Aê), {Ba), {By), {Bê), {(Ja), (C'/3), {(Je), {Ba), {D^), {By). Then I A, 

 and S_ intersect in a plane through (.7/3), {Ay), {Je), iig. 2:2, (.//3) /• . 



It also passes through the points {A'B) {a! ), {A'C) {a ) {A'B) {a\) 



^ ' II P II y II à' 



(diagram VII points 3, 1 and 2). And/y'and7cJ intersect in a plane 

 througli {A è), {B ê), {Ce), shewn in pi'ojection by the line {A S) V.,, 

 %. 20. This gives the points {fa)jD' ), {ffi) ^{B' ), {ly)jB\), 



i. e. the points (3, 5 and 4 of diagram VII. Likewise JS_ and / (./ S) in- 



■> ' 



tersect in a plane through {A' C) {a ) and parallel to the in- 



;i ."'''. 



tersection of / {A ê) and tS' . This plane is shewn in projection by 



the line F r^, fig 21; A F and ê r are equal to^; ƒ ;(^T') (//3) 

 is bisected, so that we have the points {A^)f (us' J, {A ) {«■' X (^ ) (^' ), 

 {è^)^, {B',,), {IC)^^ {a^), {A'B)^^ {a^), (//3)^^ {D' ,), (^ry),, 0,,), 

 {A'C)\_{èp), {^CB)\_{d'y). In diagram VII these points correspond to 



7, 8, 12, 11, 1, 3, 5, 4, 10, 9. Tlie synunetry shews that this is the 

 central section ; the other sections are now repeated in reverse order. 

 In practicallv constructing the sections I have found that their 

 symmetry is made more obvious by colouring the faces. The letters 

 on the faces of diagrams I — XIV denote colours, the plane and 

 accented letters denote corresponding colours and the remaining 

 sections are the same as those given with the plane and accented 

 letters interchanged. 



17. Nundjers of tetrahedra of each colour in the GOO-cell as 

 shewn by the sections. 



L and B' , 21 each, 42 



31 „ M', m each, J 20 



N „ N', 72 „ , 144 



P F' I 



R : M'\-'' ••■ '■"> 



CY CI' j 



J ;; ^. I 1^ . , 48 



X 48 4S 



Y 90 90 



Z 6 G 



Total. . 000 



