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Napoleonic triangles on the sphere

Serena Dipierro, Lyle Noakes, Enrico Valdinoci

Abstract

As is well-known, numerical experiments show that Napoleon's Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere $S^2$. Spherical triangles for which an extension of Napoleon's Theorem holds are called ``Napoleonic'', and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon's Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations.

Napoleonic triangles on the sphere

Abstract

As is well-known, numerical experiments show that Napoleon's Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere . Spherical triangles for which an extension of Napoleon's Theorem holds are called ``Napoleonic'', and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon's Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations.
Paper Structure (7 sections, 2 theorems, 52 equations, 7 figures)

This paper contains 7 sections, 2 theorems, 52 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Napoleonic questions
  3. Napoleonic examples and counterexamples
  4. Classification of Outward and Inward Napoleonic Spherical Triangles
  5. Spherical Triangles
  6. Proof of Theorem \ref{['THEOREM1']}
  7. Context

Key Result

Theorem 1.1

Given distinct $P_0$, $P_1$, $P_2\in E^2$, let $Q_0$, $Q_1$, $Q_2\in E^2$ be such that, for every $i\in \mathbb{Z}/3\mathbb{Z}$, the triangle $Q_iP_{i+1}P_{i+2}$ is equilateral with interior disjoint from the interior of the triangle $P_0P_1P_2$. Then $R_0R_1R_2$ is equilateral, where $R_i$ is the

Figures (7)

  • Figure 1: Vertices (green) of the outward Napoleonisation of the spherical triangle $P_0P_1P_2$, with $P_0:=(1,0,0)$, $P_1:=\left( \frac{3}{4},\frac{1}{4},\frac{\sqrt6}{4}\right)$ and $P_2:=\left( \frac{1}{2},\frac{1}{2},\frac{\sqrt2}{2}\right)$ (blue).
  • Figure 2: Different views of the Napoleonic construction related to \ref{['LJSD:PSL-NO0']} and \ref{['LJSD:PSL-NO1']} (the triangle $P_0P_1P_2$ being in blue and the points $R_0$, $R_1$, $R_2$ in green).
  • Figure 3: The ellipsoid of revolution $\{d_0^2+d_1^2+d_2^2+d_0d_1+d_0d_2+d_1d_2=2\}$ and its intersection with the region where the $d_i\in (0,\sqrt{3})$.
  • Figure 4: Constructing an equilateral spherical triangle $P_0P_1Q_2$, given the spherical triangle $P_0P_1P_2$.
  • Figure 5: Aerial views of Canberra, Damascus, Gramichelle, Hamina, Naples, New Dheli, Perth, Palmanova, Paris, Rome, Washington (screenshots from Google Earth).
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1.1: Napoleon's Theorem
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Example 1.6
  • Theorem 1.7