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Napoleonic Constructions in the Hyperbolic Plane

Serena Dipierro, Lyle Noakes, Enrico Valdinoci

TL;DR

This work analyzes Napoleonic constructions in the hyperbolic plane using the hyperboloid model in Minkowski space, showing that non-equilateral Napoleons cannot occur and that only equilateral triangles are Napoleonic in ℍ. It introduces bespoke hyperbolic coordinates $d_0,d_1,d_2$ and symmetric polynomials $oldsymbol{eta{=}\

Abstract

In the Euclidean setting, Napoleon's Theorem states that if one constructs an equilateral triangle on either the outside or the inside of each side of a given triangle and then connects the barycenters of those three new triangles, the resulting triangle happens to be equilateral. The case of spherical triangles has been recently shown to be different: on the sphere, besides equilateral triangles, a necessary and sufficient condition for a given triangle to enjoy the above Napoleonic property is that its congruence class should lie on a suitable surface (namely, an ellipsoid in suitable coordinates). In this article we show that the hyperbolic case is significantly different from both the Euclidean and the spherical setting. Specifically, we establish here that the hyperbolic plane does not admit any Napoleonic triangle, except the equilateral ones. Furthermore, we prove that iterated Napoleonization of any triangle causes it to become smaller and smaller, more and more equilateral and converge to a single point in the limit.

Napoleonic Constructions in the Hyperbolic Plane

TL;DR

This work analyzes Napoleonic constructions in the hyperbolic plane using the hyperboloid model in Minkowski space, showing that non-equilateral Napoleons cannot occur and that only equilateral triangles are Napoleonic in ℍ. It introduces bespoke hyperbolic coordinates and symmetric polynomials $oldsymbol{eta{=}\

Abstract

In the Euclidean setting, Napoleon's Theorem states that if one constructs an equilateral triangle on either the outside or the inside of each side of a given triangle and then connects the barycenters of those three new triangles, the resulting triangle happens to be equilateral. The case of spherical triangles has been recently shown to be different: on the sphere, besides equilateral triangles, a necessary and sufficient condition for a given triangle to enjoy the above Napoleonic property is that its congruence class should lie on a suitable surface (namely, an ellipsoid in suitable coordinates). In this article we show that the hyperbolic case is significantly different from both the Euclidean and the spherical setting. Specifically, we establish here that the hyperbolic plane does not admit any Napoleonic triangle, except the equilateral ones. Furthermore, we prove that iterated Napoleonization of any triangle causes it to become smaller and smaller, more and more equilateral and converge to a single point in the limit.

Paper Structure

This paper contains 5 sections, 13 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on the Hyperboloid Model and the Hyperbolic Triangles
  3. A Bespoke Set of Hyperbolic Coordinates
  4. Napoleonic Triangles and Proof of Theorem \ref{['non-exis']}
  5. Napoleonic Progressions and proof of Theorem \ref{['REPENE']}

Key Result

Theorem 1.1

If the Napoleonization of a hyperbolic triangle is equilateral, then the initial triangle is equilateral too.

Figures (2)

  • Figure 1.1: External Napoleonization of the Euclidean triangle with vertices in $(0,0)$, $(1,2)$ and $(3,4)$.
  • Figure 1.2: Internal Napoleonization of the Euclidean triangle with vertices in $(0,0)$, $(1,2)$ and $(3,4)$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 17 more