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Optimally Packing a Large Square by Unit Squares

Rory McClenagan

TL;DR

The paper tackles the classical square packing problem by unit squares, establishing an improved bound $W(x)=O(x^{3/5})$ on the wasted area for a square of side length $x$. It introduces a two-stage geometric packing strategy: first packing most of $S(x)$ with inclined stacks to form a trapezoidal region $T$, then applying two complementary packing algorithms within $T$ to control angular drift and error accumulation. Key insights include deriving angular relations, such as $\\theta \\asymp 1/\\sqrt{h}$ and $\\varphi^2 \\asymp 2\\theta$, and balancing parameters so that the total wasted area scales as $W(x)=O(x^{3/5})$ when $h\\sim x^{4/5}$. By carefully coordinating the two algorithms and optimizing $h$, the authors achieve the claimed bound and refine prior results in unit-square packing. This has implications for combinatorial geometry and may inform related packing and tiling problems with sharp asymptotics.

Abstract

We show that a large square of sidelength $x$ can be packed by unit squares in a manner so that the wasted space $W(x) = O(x^{3/5})$.

Optimally Packing a Large Square by Unit Squares

TL;DR

The paper tackles the classical square packing problem by unit squares, establishing an improved bound on the wasted area for a square of side length . It introduces a two-stage geometric packing strategy: first packing most of with inclined stacks to form a trapezoidal region , then applying two complementary packing algorithms within to control angular drift and error accumulation. Key insights include deriving angular relations, such as and , and balancing parameters so that the total wasted area scales as when . By carefully coordinating the two algorithms and optimizing , the authors achieve the claimed bound and refine prior results in unit-square packing. This has implications for combinatorial geometry and may inform related packing and tiling problems with sharp asymptotics.

Abstract

We show that a large square of sidelength can be packed by unit squares in a manner so that the wasted space .
Paper Structure (4 sections, 1 theorem, 42 equations, 7 figures)

This paper contains 4 sections, 1 theorem, 42 equations, 7 figures.

Table of Contents

  1. Packing S(x) Using Stacks of Unit Squares
  2. The First Packing Algorithm
  3. The Second Packing Algorithm
  4. Proof of Theorem \ref{['thm-main']}

Key Result

Theorem 1

The wasted space in packing the square $S(x)$ by unit squares is bounded by

Figures (7)

  • Figure 1: The trivial packing of $S(x)$ by squares of unit sidelength. The wasted space $W(x)$ is only $O(x)$ if $\{x\} = x - \lfloor x \rfloor$ is bounded away from $0$.
  • Figure 2: We begin by packing $S(x)$ trivially except for two rectangles of width $h$.
  • Figure 3: We pack $R$ by $n$-length parallel stacks of unit squares inclined at an angle $\theta$ such that each stack touches both the top and the bottom of $R$. We call one of the unpacked trapezoids formed $T$.
  • Figure 4: The first packing algorithm.
  • Figure 5: The second packing algorithm.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1