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Fejes Toth's solidity conjecture for the triangular packing

A solid packing of circles in the plane is one for which no finite subset of circles can be rearranged, together with the remaining circles, to form a noncongruent packing. The triangle packing is the packing by equal disks whose centers form a triangular lattice.

Fejes Toth's conjecture. The triangle packing in the plane, with one packing disk removed, is solid.

This conjecture concerns the rigidity of the triangular packing under finite rearrangements. The paper presents it as a motivation for studying finite packings in the triangular torus and gives related evidence, but does not provide a proof.

Sources & referencesView supporting material

Primary source

Robert Connelly and William Dickinson, “Periodic Planar Disk Packings”, arXiv:1201.5965 (2013).

circle-packingsrigiditytriangular-lattice

Progress summary

Refreshed
Open

The conjecture remains unresolved: the triangular arrangement with one circle removed is known to be a particularly rigid candidate, but no proof or counterexample has been found.

Fejes Toth conjectured that removing one disk from the planar triangular packing still leaves a packing that cannot be changed by any finite rearrangement. A 2012 paper uses this conjecture to motivate related finite-packing questions, but does not prove it.

Known results

  • The 2012 paper formulates a related density-gap conjecture for nn equal disks in a triangular torus, with the one-disk-removed triangular packing as the conjectured unique extremizer when n+1n+1 is a triangular lattice number and nn is not.
  • Earlier work determined all locally maximally dense packings for 1≤n≤61\le n\le 6.
  • Density-gap results were obtained for several classes of collectively jammed packings, including triangle-rhombus strip tilings, providing evidence but not a proof of the solidity conjecture.

Current status (as of August 2026): Fejes Toth’s solidity conjecture remains open; related finite-packing evidence is known, but no proof, counterexample, or verified recent resolution is recorded.

Sources

Solutions 0

No solutions have been posted yet.