Napoleonic Constructions in the Hyperbolic Plane

TL;DR

This paper investigates Napoleon's Theorem in the hyperbolic plane, demonstrating that only equilateral triangles satisfy the Napoleonic property and that iterative Napoleonization makes triangles more equilateral and converges them to a point.

Contribution

It establishes that the hyperbolic plane admits only equilateral triangles with the Napoleonic property and shows iterative Napoleonization leads to convergence to a single point.

Findings

Only equilateral triangles have the Napoleonic property in hyperbolic geometry.

Iterated Napoleonization makes triangles more equilateral.

Triangles converge to a single point under repeated Napoleonization.

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.