Grationality, With a Spoon
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866913985449689088 |
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| author | Clark, L. Jeneva |
| author_facet | Clark, L. Jeneva |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Grationality, With a Spoon Clark, L. Jeneva History and Overview 00 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. |
| title | Grationality, With a Spoon |
| topic | History and Overview 00 |
| url | https://arxiv.org/abs/2508.08267 |