Covering a Regular Tetrahedron with Diminished Copies
Fangyu Zhang
School of Mathematics, Tianjin University, China.
Yuqin Zhang
School of Mathematics, Tianjin University, China
Mei Han *
School of Mathematics, Tianjin University, China.
*Author to whom correspondence should be addressed.
Abstract
Let T be a unit regular tetrahedron. A diminished copy of T is the image of T under a homothety with positive ratio smaller than 1. Let m be a positive integer and let γm(T) be the smallest positive number r such that T can be covered by m translates of rT. Zong gave the results of γ4(T) = 3/4and γ5(T) = 9/13. However, the values of γ6(T) , γ7(T) and γ8(T) were not given then. In this article we give the upper bounds of γ6(T), γ7(T) and γ8(T).
Keywords: Covering, tetrahedron, Hadwiger’s conjecture