Lower Bound on Translative Covering Density of Tetrahedra

TL;DR

This paper establishes the first meaningful lower bound on the translative covering density of tetrahedra, demonstrating that it exceeds 1 by a small but significant margin, relative to cubes.

Contribution

It provides the first nontrivial lower bound on tetrahedral translative covering density, advancing understanding of geometric covering problems.

Findings

Lower bound on tetrahedral covering density is 1 + 1.227×10^{-3}

Bound is relative to the covering density of cubes

First nontrivial lower bound for this problem

Abstract

In this paper, we present the first nontrivial lower bound on the translative covering density of tetrahedra. To this end, we show the lower bound, in any translative covering of tetrahedra, on the density relative to a given cube. The resulting lower bound on the translative covering density of tetrahedra is $1+1.227\times10^{-3}$.