2

JOSEPH ZAKS

Another example of eight neighborly tetrahedra was given by S. Wilson and the author [17];

here, too, there are two quadruples of tetrahedra separated by a plane which contains a facet of each

one of the tetrahedra; this example is shown in Figure 2.

Figure 2. The bases of eight tetrahedra, in another example

of a neighborly family, taken from [17].

Baston stated an argument which implies that a neighborly family of tetrahedra contains at

most seventeen members. He conjectured that the maximum number of tetrahedra in a neighborly

family is eight. Baston [2] showed in 1965 that this maximum is at most nine, and he repeated

Bagemihl's conjecture that the maximum is eight.

Bagemihl's conjecture had been repeatedly mentioned in the literature: by Danzer, Grunbaum

and Klee [5] in 1963, by Grunbaum [8] in 1967 and in particular by Klee [10] in 1969 (it is

also mentioned by Klee in his research-film called "Shapes of the Future", part two, produced by

the M.A.A. at about 1972); it is also mentioned by Perles [12], by kassem [9] and by us [17,18].