NOTES ON POLYHEDRA. 
V5 — 1 . V5 + 1 j 
± T 
V5 - 1 , V5 + 1 . 
ï * —4 Î" 
V5-1 . V 5 + 1 . 
40 cube roots of ± 1 
± ”1—*±—4—.7 
± i, ± j, ± 
+ + 
Vo + i. V5 — 1 
4 
J ± 
k, 
, . . V5 + 1 7 V5 — 1 . 
± 47 + —-j— A? + —— 1, >■ 30 square „ 
4 4 
17 V5 + 1. V5 — 1. 
± P ± —* ± —4— j> 
+1 
2 terms 
120 
(The signs + are all independent, except that in each of the three expressions in the 
top line the signs in V5 + 1, V5 p 1, are opposite to each other, so that each of the 
three expressions has 16 values.) 
It is to be remarked that in the groups of 24 and 48, the group is not altered 
by any permutation whatever of the symbols i, j, k ; whereas the group of 120 is not 
altered by the cyclical permutation of these symbols, but it is altered by the inter 
change of any two of them ; the geometrical reason of this difference may be perceived 
without difficulty. 
P.S. I found accidentally, Gergonne, t. xv., p. 40, (1824—25), the following 
problem : “ De combien de manières m couleurs différentes les unes des autres peuvent- 
elles être appliquées sur les faces d’un polyèdre régulier; m représentant tour à tour 
les nombres 4, 6, 8, 12, 20?” 
Instead of the m of the problem, writing as before F for the number of faces, 
and writing also E for the number of edges; then if different positions of the same 
polyhedron were reckoned as different polyhedra, the number of ways would of course 
be II(F) (=1.2.3 ...F); and since by what precedes the same polyhedron can be 
placed in 2E positions, the required number of ways is —II (F). 
LKt 
Thus for the tetrahedron, if the colours are black, white, red, green, we may place 
it with the black face on the table and the white face in front; the only variation 
in the disposition of the colours, is according as the right hand and the left hand 
faces are coloured red and green or else green and red respectively; and the number 
of ways therefore is = 2, which agrees with the formula. 
2, Stone Buildings, W.G., 30 January, 1863. 
68—2