Circle Packing

A page of circles, each grown until it touches a neighbour or the wall.

What it is

Circles placed one at a time at random candidate positions, each inflated until it kisses an existing circle or the page boundary, until no circle of the minimum radius fits anywhere. The result is a dense, organic arrangement — bubbles, pebbles, cells — with tangencies everywhere and overlaps nowhere.

How to use

A colouring page with a built-in rhythm: big circles anchor a palette, small ones fill with accents. The optional concentric rings turn each circle into a target of bands. Kids mode raises the minimum radius so every circle is comfortably fillable with a crayon.

Purpose

The design lane’s simplest satisfying object — and the crate that demonstrates deriving a default from a rule rather than taste: the minimum radius (11 Pt) is computed from the colourability gate’s 40 mm² minimum region area, because the first default (9 Pt) made every page fail and escalate invisibly.

History

Tangent circles run from Apollonius of Perga (the tangency problem, ~200 BC) through Apollonian gaskets to Descartes’ circle theorem (1643). The grow-until-touch packing is a staple of generative art practice; circle packings also carry serious mathematics (Koebe–Andreev–Thurston: every planar graph is a circle-packing contact graph).

This implementation

• Spec knobs: size, shape (page silhouette), min_radius, max_radius, candidates per placement, rings, stroke, kids.
• Generation: rejection-sampled candidate centres; each circle takes the largest legal radius at its position, capped by max_radius; placement stops when candidates successive positions all fail at min_radius.
• Guarantees: no overlaps (checked, with the check exposed as check() -> Result), all circles inside the page shape, deterministic per seed; colourability-gated by construction since the radius floor is the region-area rule.