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

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THE REGULAR POLYHEDRON

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and, this being the case, the arc B connecting P, and Pt (the
points of P1,..., P& located on Cs and C) would have a
length <A. For, by trigonometry,
cos B = cos2 A +sin2 A cost cos22 A + sin2 A > cos A;
and, since 0<B<90~, it follows that B<A. But this is contrary to hypotheses, since the lengths of the arcs radiating
from Ps are equal to the lengths of the arcs radiating from Pi.
Let m> 2. Then it follows that there are just m arcs of
length A radiating from P1, each making an angle of (360/m)0
with its adjacent arcs. The same will be true for each of the
points P2,..., Pk, and we see readily that the sphere will
be divided by all the arcs of length A, joining the various points
P1,..., Pi which can be reached from one of them by passing along such arcs, into a number of equal and regular polygons.
Accordingly, these points, say Pi,..., Pi, are the vertices
of a regular polyhedron inscribed in 2.
Consider next the case where there are no axes of index
greater than 2. Proceeding as above, we let L denote an axis
of index 2, and we obtain the points P1,..., Pk by G'. There
are at least two arcs of length A radiating from Pi making an
angle of 180~ with each other. Taken together they form a
single arc C upon which (when extended round the sphere)
P1 and some other points P2,.., Pz lie, equally distributed
over the entire circle. If 1> 2, a rotation of 180~ around Pi
followed by a rotation of 180~ around one of the points next
to P1 is equivalent to a rotation of (720/1)~ around an axis
perpendicular to the plane of the circle C.
Every axis is of index 2 by assumption. It follows that
2/1=1 or 1/2; i.e., 1=4. In this case we have three mutually perpendicular axes of index 2.
The distance A is therefore either 180~ or 90~. In the for
mer case we have a single axis of index 2 in G'. In the latter
case there are four arcs of length A radiating from P1, lying
on two circles which are at right angles to each other at P1.
Their extremities lie in the diametral plane which is perpendicular to the axis L, and must be 90~ or 180~ apart. Con