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].
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