Square donuts and twistable holes

TL;DR

This paper characterizes when a square donut with an integral-sided hole can be rotated 90 degrees while remaining a donut, providing a full classification of such donuts, especially those with square shapes, linked to Pythagorean triples.

Contribution

It establishes necessary and sufficient conditions for rotating square donuts with integral sides and classifies all such donuts, connecting square donuts to Pythagorean triples.

Findings

Conditions for 90-degree rotation of square donuts are derived.

Complete classification of square and square-holed donuts is provided.

Square donut classification relates to Pythagorean triples.

Abstract

A mathematical donut is a rectangle of integral side length with a smaller rectangle (called the hole of the donut), also of integral side length, strictly inside it and with sides of the rectangles parallel to each other, where the area of the larger rectangle is twice that of the smaller. Necessary and sufficient conditions are determined for when the hole of the donut can be rotated $90^{\circ}$ and a donut still exists, and a complete classification of all square or square-holed donuts is given, with the square donut classification being intimately related to Pythagorean triples.