Super-regular polytopes in cyclotomic hypercubes

TL;DR

This paper demonstrates that cyclotomic hypercubes exhibit high-dimensional super-regularity properties, with most triangles nearly equilateral and pyramids nearly regular as the prime p increases.

Contribution

It establishes the super-regularity of cyclotomic hypercubes and quantifies geometric properties that emerge in high dimensions as p tends to infinity.

Findings

Almost all triangles are nearly equilateral.

Almost all angles at the origin are nearly right angles.

Most pyramids are nearly regular with nearly equal edges and nearly isosceles faces.

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.