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  in 1905
and Besicovitch  in 1947 gave an example of an infinite neighborly family of convex
In 1956, Bagemihl  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.