Dual Rectangles

TL;DR

This paper explores the duality concept in rectangles, proving finiteness of integral dual pairs, establishing a group law on rational self-dual rectangles, and using cubic surface arithmetic to generate new rational dual pairs.

Contribution

It introduces a novel duality notion for rectangles, proves finiteness of integral dual pairs, and applies elliptic curve-inspired methods to construct rational dual rectangles.

Findings

Finitely many integral dual rectangles exist.

A group law is established on rational self-dual rectangles.

New rational dual rectangles are constructed using cubic surface arithmetic.

Abstract

This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given that only finitely many integral sided pairs of dual rectangles exist. Then a geometrical group law is shown to hold on the set of all rational self-dual rectangles. Finally, the arithmetic of a cubic surface is used to construct new pairs of rational dual rectangles from old, a technique inspired by the theory of elliptic curves.