Some properties of polyhedra in Euclidean space

Abstract

The problem we consider here arises quite naturally from Crum's Problem which asks: What is the maximum number of non-overlapping polyhedra such that any pair of them have a common boundary of positive area? In (1) Besicovitch, by constructing a sequence of polyhedra satisfying the required conditions, showed that the answer to Crum's Problem is infinity. In this thesis we ask: What is the answer when the polyhedra of Crum's Problem are restricted to be tetrahedra? As stated in the Abstract, we prove that the answer is either 8 or 9 and the evidence tends to point to the answer in fact being 8. As the difference in answers would suggest, the methods we use to establish these results are completely different from those used by Besicovitch in his paper. In Chapter One we show that an n-con (for definition see the Abstract) can be represented by an n-Towed matrix whose minors satisfy certain conditional we then develop arguments from which we deduce that n is less than 18. Chapter Two shows that the bound may be reduced to n less than 14. The subsequent five chapters are mainly concerned with the conditions under which a 9-con can exist and we eventually show that if an n-con for n>9 exists then no plane contains six faces of the tetrahedra of the n-con.5.Our analysis of the 9-con continues in Chapter Eight where we show that what one may describe as the most symmetrical case for a 9-con cannot exist and also that the faces of the tetrahedra of the 9-con must be so arranged that they are contained in either nine or ten planes. To demonstrate that the existence or not of a 9-con is critical, we show in Chapter Nine that a 10-con cannot exist and in Chapter Ten that a 8-con does exist. Chapter Eleven discusses the results obtained.