Halving Lines and Underlying Graphs

In an Apollonian circle packing, the curvatures a, b, c, d of any four mutually tangent circles satisfy Descartes’ equation, 2(a + b + c + d) = (a+ b+ c+ d). For an equilateral triangle and a point P , if a, b, c, d denote the squares of the lengths of the sides of the triangle and the distances from P to the vertices of the triangle, then a, b, c, d, satisfy 3(a+ b+ c+d) = (a+ b+ c+d). Define a quadruple of nonnegative integers (a, b, c, d) to be a triangle quadruple if it satisfies this equation. It is easy to verify that if (a, b, c, d) is a triangle quadruple, then (a, b, c, a+ b+ c− d) is also. This operation and analogous ones for the other elements can be represented by four matrices, and define the triangle group to be the group with these as generators. This work analyzes some properties of triangle quadruples and the triangle group. We show that all triangle quadruples can be reduced to one unique root quadruple, and that all primitive root quadruples are contained in one orbit. We also compute the largest triangle quadruple obtainable at each stage and an approximation for the number of triangle quadruples below a given height. Finally, we prove that the triangle group is a hyperbolic Coxeter group.