A family of convex d-polytopes in the d-space is called neighborly
[1,5,8,9,10,12,14,17-22] if every pair of its members intersect in a (d-l)-dimensional set; such
an intersection lies in a hyperplane which separates the pair and contains a facet of each one of
In studying possible extensions of the Four-Color Conjecture to E3, Tietze [15] in 1905
and Besicovitch [3] in 1947 gave an example of an infinite neighborly family of convex
3-polytopes in
In 1956, Bagemihl [1] restricted the attention to neighborly families of
tetrahedra. Bagemihl gave the example of eight neighborly tetrahedra, shown here in Figure 1. All
the tetrahedra have a facet on a common plane, which separates four of them from the remaining
four. Each one of the two quadruples shares a common vertex in the open half-space determined
by the said plane.
Figure 1. The bases of eight neighborly tetrahedra. Four of the tetrahedra
are above the plane, and the remaining four are below it.
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