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

TL;DR

This paper determines the minimal total perimeter for axis-aligned rectangular tilings of a flat torus and proves that this minimum is achieved with either one or two rectangles, providing exact bounds.

Contribution

It establishes the exact minimum perimeter sum for such tilings and characterizes the configurations that attain this minimum.

Findings

Minimum perimeter sum is achieved by either one or two rectangles.

Exact minimum perimeter values are derived for axis-aligned rectangular tilings.

Characterization of tilings that attain the minimum perimeter.

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.