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No Nine Neighborly Tetrahedra Exist
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This book is devoted to a proof of the following problem of F Bagemihl (1956): What is the maximum number of tetrahedra in three-space such that every two of them meet in a two-dimensional set? Such families are called neighborly, Bagemihl presented an example of eight neighborly tetrahedra and showed that a neighborly family of tetrahedra contains at most seventeen members. This upper bound was reduced to nine in 1965. The question of whether or not there can be nine neighborly tetrahedra has been repeatedly mentioned in the literature since 1956. This book also treats the problem of the numb...
This book is devoted to a proof of the following problem of F Bagemihl (1956): What is the maximum number of tetrahedra in three-space such that every two of them meet in a two-dimensional set? Such families are called neighborly, Bagemihl presented an example of eight neighborly tetrahedra and showed that a neighborly family of tetrahedra contains at most seventeen members. This upper bound was reduced to nine in 1965. The question of whether or not there can be nine neighborly tetrahedra has been repeatedly mentioned in the literature since 1956. This book also treats the problem of the number of combinatorially different examples of eight neighborly tetrahedra. The author concludes by reproducing a proof that there can be at most fourteen tetrahedra in three-space such that every two tetrahedra are separated by a plane containing a facet of each of them. The book allows readers to follow the solution of this long-standing open problem by using various tools, including a few extensive computer searches.
Table of contents:
The Baston matrix; the 24 solutions; Cases #4, 5, 7, 8, 11, 14, 15, 19 and #24; Types of tetrahedra; The use of the Baston matrices; Computer search for case #2; Computer search for case #3; Computer search for case #6, 9 and #10; Computer search for case #12, 16, 17 and #18; Computer search for case #20, 21, 22 and #23; End of the proof; On eight neighborly tetrahedra; At most fourteen nearly-neighborly tetrahedra.
Table of contents:
The Baston matrix; the 24 solutions; Cases #4, 5, 7, 8, 11, 14, 15, 19 and #24; Types of tetrahedra; The use of the Baston matrices; Computer search for case #2; Computer search for case #3; Computer search for case #6, 9 and #10; Computer search for case #12, 16, 17 and #18; Computer search for case #20, 21, 22 and #23; End of the proof; On eight neighborly tetrahedra; At most fourteen nearly-neighborly tetrahedra.