On Decomposition of Embedded Prismatoids in $\mathbb {R}^3$ Without Additional Points

Abstract

This paper considers embedded three-dimensional prismatoids. A subclass of this family is twisted prisms, which includes the family of non-triangulable Schönhardt polyhedra (Schönhardt, Math Ann 98:309–312, 1928; Rambau, On a generalization of Schönhardt’s polyhedron. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, vol. 52, pp. 501–516. MSRI Publications, Chicago, 2005). We call a prismatoid decomposable if it can be cut into two smaller prismatoids (which have smaller volumes) without using additional points. Otherwise, it is indecomposable. The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa. In this paper, we prove two basic facts about the decomposability of embedded prismatoid in $\mathbb {R}^3$ with convex bases. Let P be such a prismatoid, call an edge interior edge of P if its both endpoints are vertices of P, and its interior lies inside P. Our first result is a condition to characterize indecomposable twisted prisms. It states that a twisted prism is indecomposable without additional points if and only if it allows no interior edge. Our second result shows that any embedded prismatoid in $\mathbb {R}^3$ with convex base polygons can be decomposed into the union of two sets (one of them may be empty): a set of tetrahedra and a set of indecomposable twisted prisms, such that all elements in these two sets have disjoint interiors.