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Uniqueness of Optimal Point Sets Determining Two Distinct Triangles

Hazel N. Brenner, James S. Depret-Guillaume, Eyvindur A. Palsson, Steven Senger

Abstract

In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the work of Hirasaka and Shinohara who determined that the $d$-orthoplex is such an optimal configuration, but did not prove its uniqueness. Further, we present a more elementary argument for its optimality.

Uniqueness of Optimal Point Sets Determining Two Distinct Triangles

Abstract

In this paper, we show that the maximum number of points in dimensions determining exactly 2 distinct triangles is . We further show that this maximum is uniquely achieved by the vertices of the -orthoplex. We build upon the work of Hirasaka and Shinohara who determined that the -orthoplex is such an optimal configuration, but did not prove its uniqueness. Further, we present a more elementary argument for its optimality.

Paper Structure

This paper contains 7 sections, 12 theorems, 6 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions, Lemmas and Setup
  3. Eliminating Higher Numbers of Distinct Distances
  4. Geometry of Two Distinct Triangles Determined by 2d Points
  5. Proving the Main Result
  6. Proofs of Lemmas
  7. Some Notes on Elementary Euclidean Geometry

Key Result

Theorem 1.1

The vertices of the $d$-orthoplex are the unique optimal configuration determining two distinct triangles in $\mathbb{R}^d$ for $d\geq 3$.

Figures (5)

  • Figure 1: Maximal configurations determining exactly $k$ distances, for $2 \leq k \leq 6$brass_et_al. Note that for each $k>2$, there is an example from the triangular lattice; it is conjectured that this is always the case for $k$ large enough.
  • Figure 2: The distinct distances present are $d_1, d_2$ and $d_3$, so $n=3$. Of these, $d_2$ and $d_3$ are repeated, so $m=2$. Thus, this configuration determines at least 3 distinct triangles.
  • Figure 5: The setup for the 3-dimensional version of this argument
  • Figure 6: In the 3-dimensional case, our simplex is the tetrahedron whose vertices are $\mathcal{O}$, $A$, $B$ and $C$. The point $D$ is the center of the tetrahedron. It is intuitively clear from this image that there is no consistent position for the contradictory point $E$ other than $D$.
  • Figure 7:

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.3
  • Lemma 2.3
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • ...and 19 more