Question 1

What is an Apollonian gasket?

Accepted Answer

An Apollonian gasket is a fractal created by starting with three mutually tangent circles and repeatedly filling every curvilinear triangular gap with the largest circle that fits — the Soddy circle. It is named after Apollonius of Perga, who studied the problem of constructing circles tangent to three given circles (~200 BC). Apollonius of Perga's own works were lost; the gasket construction stems from Leibniz and Soddy (1936).

Question 2

What is Descartes' Circle Theorem?

Accepted Answer

Descartes' theorem (rediscovered 1643) states: if four mutually tangent circles have curvatures (bends) k₁, k₂, k₃, k₄ = 1/r, then (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²). For the outer enclosing circle, its curvature is negative. Given k₁,k₂,k₃, the two Soddy circles have curvature k₄ = k₁+k₂+k₃ ± 2√(k₁k₂+k₂k₃+k₃k₁). The inverse formula k₄' = 2(k₁+k₂+k₃)−k₄ (which avoids square roots) is used for recursive generation once the initial packing is known.

Question 3

Why do integer curvatures appear in Apollonian gaskets?

Accepted Answer

Starting from an initial packing with integer curvatures (like −1, 2, 2, 3), every subsequently generated circle also has integer curvature. This follows from the inverse Descartes formula k₄' = 2(k₁+k₂+k₃)−k₄: since k₁,k₂,k₃,k₄ start as integers, all derived curvatures remain integers. This remarkable arithmetic property connects Apollonian gaskets to number theory — specifically to the representation of integers by the Apollonian binary quadrilateral form.

Question 4

What is the fractal dimension of an Apollonian gasket?

Accepted Answer

The Apollonian gasket has a Hausdorff dimension of approximately 1.3057, computed numerically by Boyd (1973) and later confirmed analytically. The residual set (everything not covered by any circle) has this dimension, meaning it is strictly between 1 (curve) and 2 (filled area). The total area of all the circles approaches the area of the outer circle — the packing is asymptotically perfect — yet the uncovered Cantor-like set has positive Hausdorff measure.

🧪 Apollonian Gasket — Fractal Circle Packing via Descartes' Theorem

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