(4) Precisely one of the three inequalities changes whenever a line between
sextants of the hexagon is crossed (see Figure 6(a)).
So we begin our search for the 3-dimensional analogue by observing
that for the trinomial coefficients
(1') There are precisely 24-2=14 regions in space, since it
is possible to have all combinations of assignments of signs to
n, r, s, t
except
n ³ 0,
r < 0,
s < 0,
t < 0
and
n < 0,
r ³ 0,
s
³ 0,
t ³ 0.
Now, what we need is a symmetric polyhedron whose extended face planes
produce 14 distinct regions in space satisfying the analogue of (2) in a
satisfactory way. So we ask first: What is the 3-dimensional analogue of
an equilateral triangle? The natural answer is the regular tetrahedron (and
we know that, when
n,r,s,t ³ 0,
the trinomial
coefficients can be arranged in a regular tetrahedral region as illustrated
in Figure 5). Now if we extend the face planes of a regular tetrahedron we
obtain
(a) 4 trihedral regions off the vertices,
(b) 4 truncated trihedral regions off the faces,
and
(c) 6 wedges off the edges (with square cross sections).