Circles of Apollonius

I have decided to investigate some mathematical problems from the ancient world. One of them is known as the "Apollonian Fractal", or the "Circles of Apollonius". Named after the Greek mathematician Apollonius of Perga, the fractal is a beautiful array of infinite circles within circles, shrinking exponentially smaller with each iteration.

One metric of a fractal is the Hausdorff Dimension, which is a measurement of a shape's roughness that was introduced in 1918. The Hausdorff Dimension of a classic Apollonian Fractal is equal to approximately 1.3056867, but is very difficult to calculate, as it requires a great amount of computing power to simulate the Fractal's design.

My program stumbled upon a single equation for the Hausdorff Dimension of a traditional Apollonian Fractal. As with the others, there is no real way to tell if it is accurate, as the mathematical proof would be prohibitively difficult. The equation is:

Apollonian Hausdorff Dimension ~= tan(((Pi)^(Pi))/10 - cosh(SQR))

{Pi = 3.14159265358979323846264338327950288; SQR = 1.66168794963359412129581892274995074}

Using mathematical notation:

ε ~= tan(^ππ/₁₀ - cosh(σ))

{ε = Apollonian Hausdorff Dimension; π = Pi; σ = Somos' Quadratic Recurrence}

ε ~= 1.305686680971073695647339380382581378988748138480994883472