On the Multiple Packing Densities of Triangles

TL;DR

This paper proves that for triangles, the maximum density of k-fold translative packings equals that of k-fold lattice packings, confirming a conjecture for this specific shape.

Contribution

It establishes the equality of k-fold translative and lattice packing densities for triangles, extending understanding of packing densities for convex shapes.

Findings

elta_T^k(T) = elta_L^k(T) for triangles

Confirms a conjecture relating translative and lattice packings for triangles

Provides explicit formulas for k-fold packing densities of triangles

Abstract

Given a convex disk $K$ and a positive integer $k$, let $\delta_T^k(K)$ and $\delta_L^k(K)$ denote the $k$-fold translative packing density and the $k$-fold lattice packing density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $\delta_L^k(T)=\frac{2k^2}{2k+1}$. In this paper, I will show that $\delta_T^k(T)=\delta_L^k(T)$.