Super-regular polytopes in cyclotomic hypercubes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929550208794624 |
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| author | Cobeli, Cristian Zaharescu, Alexandru |
| author_facet | Cobeli, Cristian Zaharescu, Alexandru |
| contents | For any odd prime $p$ and any integer $N\ge 0$, let $\mathcal{V}(p,N)$ be the set of vertices of the cyclotomic box $\mathscr{B} = \mathscr{B}(p,N)$ of edge size $2N$ and centered at the origin $O$ of the ring of integers $\mathbb{Z}[ω]$ of the cyclotomic field $\mathbb{Q}(ω)$, where $ω=\exp\big(\frac{2πi}{p}\big)$. Cyclotomic boxes represented as sets of points in the complex plane prove to have counter-intuitive super-regularity properties that are known to occur in high dimensional real hypercubes.
Employing the naturally induced Euclidean-trace metric for distance measurement and letting the prime $p$ tend to infinity, we prove the following results. 1. Almost all triangles with vertices in $\mathcal{V}(p,N)$ are almost equilateral. 2. Almost all angles $\angle VOA$, where $V$ is in $\mathcal{V}(p,N)$, $O$ is the origin, which coincides with the center of $\mathscr{B}(p,N)$, and $A$ is fixed anywhere in $\mathscr{B}(p,N)$, are right angles. 3. Almost all pyramids with base on $\mathcal{V}(p,N)$ and the apex fixed anywhere in $\mathscr{B}(p,N)$ are super-regular, meaning that the base has all edges and diagonals almost equal and the lateral faces are nearly isosceles triangles, each nearly equal to the others. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Super-regular polytopes in cyclotomic hypercubes Cobeli, Cristian Zaharescu, Alexandru Number Theory Combinatorics Metric Geometry Primary 11B99, Secondary 11P21, 52B11 For any odd prime $p$ and any integer $N\ge 0$, let $\mathcal{V}(p,N)$ be the set of vertices of the cyclotomic box $\mathscr{B} = \mathscr{B}(p,N)$ of edge size $2N$ and centered at the origin $O$ of the ring of integers $\mathbb{Z}[ω]$ of the cyclotomic field $\mathbb{Q}(ω)$, where $ω=\exp\big(\frac{2πi}{p}\big)$. Cyclotomic boxes represented as sets of points in the complex plane prove to have counter-intuitive super-regularity properties that are known to occur in high dimensional real hypercubes. Employing the naturally induced Euclidean-trace metric for distance measurement and letting the prime $p$ tend to infinity, we prove the following results. 1. Almost all triangles with vertices in $\mathcal{V}(p,N)$ are almost equilateral. 2. Almost all angles $\angle VOA$, where $V$ is in $\mathcal{V}(p,N)$, $O$ is the origin, which coincides with the center of $\mathscr{B}(p,N)$, and $A$ is fixed anywhere in $\mathscr{B}(p,N)$, are right angles. 3. Almost all pyramids with base on $\mathcal{V}(p,N)$ and the apex fixed anywhere in $\mathscr{B}(p,N)$ are super-regular, meaning that the base has all edges and diagonals almost equal and the lateral faces are nearly isosceles triangles, each nearly equal to the others. |
| title | Super-regular polytopes in cyclotomic hypercubes |
| topic | Number Theory Combinatorics Metric Geometry Primary 11B99, Secondary 11P21, 52B11 |
| url | https://arxiv.org/abs/2410.14473 |