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The minimum length of an axis-aligned rectangular tiling of a flat torus

Hau-Yi Lin, Wu-Hsiung Lin, Gerard Jennhwa Chang

Abstract

A flat torus is the quotient of the Euclidean plane over a lattice generated by a basis, and an axis-aligned rectangular tiling of a flat torus is a partition into finitely many rectangles whose sides are axis-aligned. We provide the minimum sum of the perimeter of rectangles for an axis-aligned rectangular tiling, and prove that it is attainable by either exactly one rectangle or exactly two rectangles.

The minimum length of an axis-aligned rectangular tiling of a flat torus

Abstract

A flat torus is the quotient of the Euclidean plane over a lattice generated by a basis, and an axis-aligned rectangular tiling of a flat torus is a partition into finitely many rectangles whose sides are axis-aligned. We provide the minimum sum of the perimeter of rectangles for an axis-aligned rectangular tiling, and prove that it is attainable by either exactly one rectangle or exactly two rectangles.
Paper Structure (10 sections, 9 theorems, 13 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Axis-aligned rectangular tiling of a flat torus
  4. Exactly one rectangle
  5. Exactly two rectangles
  6. Lower bound on the length of an axis-aligned rectangular tiling
  7. Skeletons that contain a horizontal/vertical cycle
  8. Skeletons that contain no horizontal/vertical cycle
  9. Minimum length of an axis-aligned rectangular tiling
  10. Computation remarks

Key Result

Theorem 2.1

Let $\Lambda$ be a lattice. If $C$ is a convex centrally symmetric region in $\mathbb{R}^2$ with volume greater than $4 d(\Lambda)$, then $C$ contains a nonzero lattice point.

Figures (1)

  • Figure 1: Two possible cases for the rectangular tiling with exactly two rectangles.

Theorems & Definitions (19)

  • Theorem 2.1: Minkowski's convex body theorem Cassels1959
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • ...and 9 more