Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 102] THE TETRAHEDRON AND OCTAHEDRON

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102. The Tetrahedron and Octahedron. We now examine
the five ordinary regular solids. Of these, the hexahedron and
octahedron furnish the same set of axes of rotation, as do also
the dodecahedron and icosahedron. We therefore have only
three cases to consider: the tetrahedron, octahedron and icosahedron.
In the case of the tetrahedron we have four vertices and
correspondingly four axes of rotation of index 3; besides, three
axes of index 2, each passing through the middle points of a
pair of opposite edges. The latter axes are mutually perpendicular and may be taken as the X-, Y-, and Z-axes. The
corresponding transformations of G are then as follows:.O i 0 1
T = (i, -i), T2=, T3 =,
i 0. -1 0
either directly or after multiplication by E1=(-l, -1). If
the vertices are named a, b, c, d, the three rotations permute
them among themselves according to the substitutions
(ab) (cd), (ad) (be), (ac) (bd).
The remaining rotations permute the vertices three at a time
cyclically, as (abc),... The corresponding transformations
of G may be determined analytically from the conditions that
they are each of order 3 and transform the collineations corresponding to T1, T2, T3 cyclically. Certain ambiguities arise
from the fact that the similarity-transformation E =(-1,
-1) is present in the group. Thus, S=(abc) may transform
T1 into T2 or into T2E1, etc. For (abc) we find four forms
possible, all of which are present in G if one of them is. We
shall choose the following form:
-1+i -1+i
2 2
S=
1+i -1-i
2 2