Grationality, With a Spoon

I introduced grationality at a 2025 sectional meeting of the Mathematical Association of America to install a handle on a concept akin to rationality of numbers, but in a geometric context. More specifically, we define a nice n-gon to be a regular n-gon with side lengths that are natural numbers; then a number n is 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. In this paper I offer 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.