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Periodic planar disc packings.

Robert Connelly1, William Dickinson

  • 1Department of Mathematics, Cornell University, , Ithaca, NY 14853, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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PubMed
Summary
This summary is machine-generated.

This study explores conditions for maximally dense disc packings in a torus. It proposes a density limit for jammed packings, excluding triangle-only configurations, which holds true for several packing classes.

Keywords:
collectively jammedlatticepacking densityperiodic packingrigidtriangle and square tiling

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Area of Science:

  • Geometry
  • Discrete Mathematics
  • Materials Science

Background:

  • Investigates the geometric problem of packing equal discs within a torus, a space defined by a 2D lattice.
  • Examines conditions for locally maximal density in such disc packings.

Purpose of the Study:

  • To present conjectures on the upper bound density for collectively jammed disc packings in a torus.
  • To identify specific conditions and packing configurations where these density conjectures apply.

Main Methods:

  • Analysis of disc packing configurations within a toroidal space.
  • Focus on collectively jammed packings and their associated graphs.
  • Mathematical formulation of density limits based on the number of discs (n).

Main Results:

  • Identified several conditions under which disc packings in a torus achieve local maximal density.
  • Proposed a conjecture that density is bounded by (n/(n + 1))π√12 for non-triangular jammed packings on a triangular lattice.
  • Demonstrated that this conjecture holds for several specific classes of collectively jammed packings.

Conclusions:

  • The study provides insights into the density limits of disc packings in toroidal geometries.
  • The findings are relevant to understanding jamming phenomena and optimal packing strategies in constrained spaces.