Apollonian packings

Apollonian packings are a fascinating way to fill space with circles. Imagine starting with three touching circles; by finding the largest circle that can fit in the gap they create, you can keep adding more circles within the spaces left by existing ones. This process generates an intricate pattern with smaller and smaller circles fitting perfectly into the gaps. Apollonian packings reveal deep mathematical properties and relationships, showcasing how shapes can fill spaces efficiently and beautifully, while also linking to number theory and fractals. They illustrate the balance between order and complexity in mathematics.