Introduction

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

them.

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

E3.

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.

1