Super-regular polytopes in cyclotomic hypercubes

Abstract

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 $\mathbb{Z}[\omega]$ of the cyclotomic field $\mathbb{Q}(\omega)$, where $\omega=\exp\big(\frac{2\pi i}{p}\big)$. 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.