Pessimal packing shapes

TL;DR

This paper investigates shapes that minimize packing density, focusing on local minima in various dimensions, and proves new results about the regular heptagon and the ball's packing properties.

Contribution

It introduces new theoretical results on local minima of packing densities for convex shapes in two and three dimensions.

Findings

Regular heptagon is a local minimum in 2D.

Directional derivative of packing density at the ball is positive in all directions.

Focus on local minima due to the complexity of global minima.

Abstract

We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions, this question is expected to have a different answer or perhaps no answer at all. As the problem of identifying global minima in most cases appears to be beyond current reach, in this paper we focus on local minima. We review some known results and prove these new results: in two dimensions, the regular heptagon is a local minimum of the double-lattice packing density, and in three dimensions, the directional derivative (in the sense of Minkowski addition) of the double-lattice packing density at the point in the space of shapes corresponding to the ball is in every direction positive.