Apollonian Equilateral Triangles

TL;DR

This paper explores the properties and classifications of quadruples related to equilateral triangles, introducing a group-theoretic framework, and extends the analysis to higher dimensions, revealing connections to hyperbolic Coxeter groups and Lie groups.

Contribution

It introduces the concept of triangle quadruples, analyzes the triangle group actions, classifies orbits, and generalizes the problem to higher dimensions with connections to advanced algebraic structures.

Findings

The triangle group is a hyperbolic Coxeter group.

Complete classification of orbits under the triangle group.

Derived formulas for counting generated quadruples.

Abstract

Given an equilateral triangle with $a$ the square of its side length and a point in its plane with $b$, $c$, $d$ the squares of the distances from the point to the vertices of the triangle, it can be computed that $a$, $b$, $c$, $d$ satisfy $3(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$. This paper derives properties of quadruples of nonnegative integers $(a,\, b,\, c,\, d)$, called triangle quadruples, satisfying this equation. It is easy to verify that the operation generating $(a,\, b,\, c,\, a+b+c-d)$ from $(a,\, b,\, c,\, d)$ preserves this feature and that it and analogous ones for the other elements can be represented by four matrices. We examine in detail the triangle group, the group with these operations as generators, and completely classify the orbits of quadruples with respect to the triangle group action. We also compute the number of triangle quadruples generated after a certain number of operations and approximate the number of quadruples bounded by characteristics such as the maximal element. Finally, we prove that the triangle group is a hyperbolic Coxeter group and derive information about the elements of triangle quadruples by invoking Lie groups. We also generalize the problem to higher dimensions.