The Apollonian circle packing theorem

The Apollonian circle packing theorem states that you can fill a plane with an infinite arrangement of tangent circles, such that every circle touches its neighbors without overlapping. Starting with three mutually tangent circles, you can iteratively insert new circles in the gaps, where each new circle is tangent to three existing ones. This process creates a dense, fractal-like pattern of circles with decreasing sizes, covering the plane completely in the limit. The theorem highlights how intricate and beautiful geometric structures can emerge from simple rules of tangency and iterative growth.