Circle packing

Circle packing is the arrangement of circles in an area so that none of them overlap. In mathematics it is a question about how tightly circles can fit; in generative art it is a way of filling a sheet with shapes that are spread out evenly without lying on a grid.¹

The densest packing

If all the circles are the same size, the tightest arrangement is hexagonal: every circle touches six others, as in a honeycomb. The circles then cover π/(2√3), about 90.69 per cent, of the plane. Joseph-Louis Lagrange proved in 1773 that this is the densest packing among regular, lattice arrangements; Axel Thue published a proof in 1890 that it is the densest of all, and László Fejes Tóth gave the first rigorous proof in 1942.¹

Circles in the gaps

With circles of different sizes, the gaps can be filled again and again. The classic case is the Apollonian gasket. Start with three circles that touch each other. In every gap between three touching circles, place a new circle that touches all three, and repeat for ever.²

The finished gasket is a fractal: the circles never cover the whole area, and what is left has a dimension of about 1.31.²

The random method

A simpler way is trial and error, with no geometry of tangent circles. The artist Tyler Hobbs describes his method in "A Randomized Approach to Circle Packing" (2016):³

1. Choose the sizes of circle you want, and roughly how many of each.
2. Start with the largest. Pick a random position and check whether the circle fits: it overlaps another circle if the distance between their centres is not larger than the two radii added together.
3. If it fits, keep it; if not, try another position. Hobbs gives up on a size after 2,000 failed attempts in a row, a balance, he writes, between speed and how dense the result becomes.
4. Move on to the next size, ending with the smallest.

Because the large circles are placed first, the small ones end up in the gaps between them.

New drawing

As a tool

A packing is also a way of choosing points. The centres of the circles are spread evenly but irregularly, which is what Poisson disc sampling also aims for. Hobbs lists circle packing as one way of choosing where curves start in a flow field: "things are pretty evenly spaced out".⁴ On a plotter, each circle can be filled with lines, as with the concentric rings in the drawing at the top of this page; see line hatching and Plotter Art.