Grationality, With a Spoon

L. Jeneva Clark

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arXiv · 2508.08267

Grationality, With a Spoon

L. Jeneva Clark

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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.

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BibTeX RIS

L. Jeneva Clark. 2025-07-28. Grationality, With a Spoon. https://arxiv.org/abs/2508.08267

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