Grationality, With a Spoon

TL;DR

This paper introduces the concept of grationality, a geometric property of numbers related to regular polygons with natural side lengths, exploring its properties through simple geometric methods and visual reasoning.

Contribution

It defines grationality in a geometric context and establishes foundational theorems about its relation to divisibility using elementary geometric techniques.

Findings

Examples of grational and non-grational numbers are provided.

Theorems relate grationality to divisibility properties.

Geometric proofs avoid complex tools, emphasizing visual reasoning.

Abstract

The introduction of Grationality at a 2025 sectional meeting of the Mathematical Association of America installed a handle on a concept akin to rationality of numbers, but in a geometric context. A nice $n$-gon was defined to be a regular $n$-gon with side lengths that are natural numbers, and a number $n$ was defined to be grational if and only if there exists a nice $n$-gon such that its area equals the sum of areas of $n$ congruent nice $n$-gons. This paper shows several examples of grational and non-grational numbers, followed by theorems about how the grationality of a number relates to its divisibility. Proofs of these theorems do not use high-powered tools, but rely on geometric constructions, proportional reasoning, tiling, dissection, the Carpets Theorem, and proof by descent. In keeping with this simplicity, a.k.a. "doing math with a spoon," images are heavily leveraged. The benefits of choosing simplistic tools are discussed.