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Super-regular polytopes in cyclotomic hypercubes

Cristian Cobeli, Alexandru Zaharescu

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 $\mathbb{Z}[ω]$ 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.

Super-regular polytopes in cyclotomic hypercubes

Abstract

For any odd prime and any integer , let be the set of vertices of the cyclotomic box of edge size and centered at the origin of the ring of integers of the cyclotomic field , where . 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 tend to infinity, we prove the following results. 1. Almost all triangles with vertices in are almost equilateral. 2. Almost all angles , where is in , is the origin, which coincides with the center of , and is fixed anywhere in , are right angles. 3. Almost all pyramids with base on and the apex fixed anywhere in 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.

Paper Structure

This paper contains 30 sections, 18 theorems, 119 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The main results
  3. Pyramids with leveled heights
  4. The norm, the distance and the normalized distance
  5. The norm
  6. The distance
  7. The central hypercube and the Euclidean norm
  8. The key formula for the norm
  9. The Euclidean diameter of the box
  10. The normalized distance
  11. On the distances from an arbitrary point to the vertices
  12. The average distance from a point to the vertices
  13. A moment about the mean
  14. Multinomial sums on vertices that cancel out
  15. The estimation of SigmaM'
  16. ...and 15 more sections

Key Result

Theorem 1

For any $\varepsilon>0$, there exists a prime $p_\varepsilon$ such that for all primes $p\geq p_\varepsilon$, for all integers $N\geq 1$, and for every point $\alpha\in\mathscr B(p,N)$, the proportion of triangles $(\beta_1,\alpha,\beta_2)$ such that where $\beta_1,\beta_2\in\mathcal{V}(p,N)$, is greater than or equal to $1-\varepsilon$.

Figures (3)

  • Figure 1: In the representation on the left-side, there are twenty-six random triangles with vertices in $\mathcal{V}(7,2)$, and on the image on the right-side there are ten pyramids with triangular bases with vertices in $\mathcal{V}(11,1)$ and the apex in $\mathscr B(11,1)$. The apexes of the pyramids and the vertices in $\mathcal{V}(p,N)$ are shown as thicker in size compared to the rest of points in $\mathscr B(p,N)$, and the vertices in $\mathcal{V}(p,N)$ are also colored differently (blue instead of red, in the electronic version).
  • Figure 2: The set of vertices $\mathcal{V}{(q,1)}$, their encircling circle and the poles highlighted for $q=13,14,15,16$.
  • Figure 3: Distinguished types of triangles $\triangle (*XY)$, where points $X$ and $Y$ are placed at a distance $D$ from each other in a central symmetric position of the square $\Box{[t]{}}_{L}$ of side $L>D$, and $*$ stands for any point in $\Box{[t]{}}_{L}$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Theorem 4
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • ...and 16 more