Circle Packing Algorithms: RSA vs Advancing Front (2D)

Algorithm

Random Sequential Addition (RSA) Advancing front (tangent-seeking)

Circle sizes

Monodisperse (one size) Polydisperse (size range)

Radius 0.045

Growth

Attempts / frame 25

Live Readout

Circles placed

Packing fraction

0.0%

Failed streak

Status

Growing…

How it works

Both algorithms fill a fixed square with non-overlapping circles and report the true measured packing fraction (total circle area ÷ container area) — not a lookup table. RSA drops a circle at a uniformly random position and keeps it only if it clears every existing circle; when even a fixed budget of random tries in a row all fail, the pack is jammed — for equal circles this saturates near the known RSA jamming density of about 54.7%, well below crystalline order. Advancing front instead samples candidate centres tangent to an existing circle's edge at a random angle and keeps the candidate touching the most neighbours with the least gap, so new circles nucleate directly against the growing cluster instead of scattering randomly — this consistently reaches a higher density than RSA because contacts are chosen, not accidental.

Monodisperse — every circle the same radius; RSA saturates near 0.547, advancing front climbs higher but stays below the proven hexagonal optimum of 0.9069.
Polydisperse — radii drawn uniformly from a range; smaller circles can still slot into gaps left by larger ones after the pack would otherwise have jammed, which is why real granular and powder packings usually beat the monodisperse limit.

RSA jamming limit ≈ 54.7% Hexagonal optimum ≈ 90.69%