On The Rupert Property Of Polyhedra

In 3-dimemsional Euclidean space R~3,a convex body P ? R~3 is a compact convex set with non-empty interior.It is said that a convex body P has Rupert property,if there is a large enough straight hole can be dug on P so that another convex body identical to P can pass through the hole.Let p be a finite point set in Rd,The convex hull of P is called a convex polytope.If all vertices of a convex polytope P lie in two parallel planes,then P is called a prismatoid.It has been proved that all Platonic solids,9 of the 13 Archimedean solids,namely,the truncated cube,the cuboctahedron,the truncated octahedron,the rhombicubocta-hedron,the icosidodecahedron,the truncated cuboctahedron,the truncated icosahedron,the truncated dodecahedron,and the truncated tetrahedronhave have Rupert property.In Chapter 1,we study the Rupert property of cones,and prove that all pyramids and cones with a circular base enjoy Rupert property.In Chapter 2,we investigate the Rupert property of prisms,We first prove that all regular prisms have Rupert property,and then generalize the conclusion to all straight prisms.Furthermore,we present some sufficient conditions for a prismatoid to have Rupert property.This paper attempts to study the Rupert property of any convex polyhedron by studying the cone,prismatoid,adding points layer by layer,and then gives a method to explore the conjecture that all convex polyhedra have Rupert property in R~3.