Vol. 57, No. 1, 1975

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ISSN: 0030-8730
Generalized combinatorial cells and facet splitting

David Wilmot Barnette

Vol. 57 (1975), No. 1, 33–45

The similarity between triangulations of the sphere and simplicial polytopes makes cells with triangulated boundaries natural generalizations of simplicial polytopes. In this paper we extend this generalization to cells whose boundaries are broken up into more general structures than just simplices. These structures are called gcc’s. In doing so we get a generalization of the d-polytope. We shall investigate a method of constructing these structures, called facet splitting. We show that almost all d-gcc’s with up to 3 + d facets can be constructed by facet splitting, and we construct a simple 4-gcc with 10 facets that cannot be constructed in this way.

Mathematical Subject Classification
Primary: 52A25
Secondary: 57C55
Received: 30 April 1974
Published: 1 March 1975
David Wilmot Barnette