Almost Squaring the Square: Optimal Packings for Non-Decomposable Squares

Abstract

We consider the problem of finding the minimum uncovered area (trim loss) when tiling nonoverlapping distinct integer-sided squares in an N × N square container such that the squares are placed with their edges parallel to those of the container. We find such trim losses and associated optimal packings for all container sizes N from 1 to 101, through an independently developed adaptation of Ian Gambini's enumerative algorithm. The results were published as a new sequence to The On-Line Encyclopedia of Integer Sequences ® . These are the first known results for optimal packings in non-decomposable squares.

A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms

“…In most publications on sphere packing, spheres are defined by the Euclidean norm. However, many applied and theoretical packing problems, e.g., producing square, hexagonal or dodecagonal CMS sensors [34] or tiling non-overlapping distinct squares in a square container [35], can be considered as sphere packing for spheres defined in a suitable norm. To the best of our knowledge, using non-Euclidean norms to define distances in sphere packing problems was first proposed in [36][37][38].…”

“…Irregular packing problems typically require special sophisticated modeling approaches and techniques to represent placement (non-overlapping, containment) conditions (see, e.g., [24] and the references therein). However, in many applications, the shapes involved are not spherical and possess similar properties, e.g., have certain levels of central symmetry [34,35]. Our objective in this paper is to describe and investigate a class of irregular packing problems where placement conditions can be stated as simple, as in sphere packing.…”