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Square packing with $O(x^{0.6})$ wasted area

Hong Duc Bui

Abstract

We show a new construction for square packing, and prove that it is more efficient than previous results.

Square packing with $O(x^{0.6})$ wasted area

Abstract

We show a new construction for square packing, and prove that it is more efficient than previous results.

Paper Structure

This paper contains 23 sections, 15 theorems, 9 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Note on independent discovery
  3. Summary of Existing Techniques
  4. A Primitive Tightly-packed Quadrilateral
  5. Description of the Packing Method
  6. Initial Configuration
  7. Modification of the Packing
  8. Analysis of the Wasted Area
  9. Lower Bound on the Wasted Area
  10. Packing a Right Trapezoid
  11. Details of the Packing Method
  12. Analysis of the Wasted Area
  13. Reduction from Packing a Square to Packing a Right Trapezoid
  14. Discussion
  15. Symmetry of the Construction
  16. ...and 8 more sections

Key Result

Lemma 1

For every $(i, j)$, square $S_{i, j+1}$ is $\Delta_1$ to the right of square $S_{i, j}$, and square $S_{i+1,j}$ is $\Delta_2$ to the right and $\Delta_3$ below $S_{i,j}$.

Figures (20)

  • Figure 1: Trivial packing method of the space between two parallel lines.
  • Figure 2: Improved packing method of the space between two parallel lines, which reduces the wasted area to $O(x^{-1/2})$ per unit distance.
  • Figure 3: First step in the packing method.
  • Figure 4: Illustration of $\Delta_1$, $\Delta_2$ and $\Delta_3$.
  • Figure 5: Calculation of $\Delta_1$, $\Delta_2$ and $\Delta_3$ from $\theta$ and $\sigma_1$.
  • ...and 15 more figures

Theorems & Definitions (25)

  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 15 more