Corner-generated circle packing

Choose the radii.
Search the gaps.

Choose integer bends b and place circles of radius 1/b inside the unit disk. Every new circle is pinned to two sides of an exposed circular corner; the search succeeds when every requested bend appears and every circle is held by contacts surrounding it.

Search input

Define the tree

Allowed bends

The bend is the curvature b = 1/r. Enter comma or space separated integers ≥ 2. Try or search .

Saved packings

Circle horizon

Node limit

Move grammar

Seed one boundary-tangent circle. Then choose an oriented contact and a permitted bend. The left-hand tangent placement becomes a child unless it overlaps. Rotated and reflected copies share one branch.

Current geometric state

Ready to search

ready

Nodes: 0
Depth: 0
Open states: 0
Dead placements: 0
Symmetry skips: 0

What this page decides

A precise search inside a deliberately restricted grammar.

A green result is a directly checkable packing certificate: interiors are disjoint, every requested bend occurs, and every small circle has at least three tangencies that do not all lie on one side of a diameter. The outer boundary counts as one contact.

An exhausted result only rules out configurations generated by these corner insertions up to the chosen circle horizon. It is not a proof that no unrestricted packing exists.