
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 -gon was defined to be a regular -gon with side lengths that are natural numbers, and a number was defined to be grational if and only if there exists a nice -gon such that its area equals the sum of areas of congruent nice -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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematics and Applications · History and Theory of Mathematics · Art, Technology, and Culture
