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Packing spheres in high dimensions with moderate computational effort.

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Researchers generated high-density sphere packings in up to 22 dimensions using the RRR algorithm. Findings suggest densities can exceed Ball

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Area of Science:

  • Geometry
  • Computational Mathematics
  • Materials Science

Background:

  • Sphere packing is a fundamental problem in geometry and discrete mathematics.
  • Existing bounds, like Ball's lower bound and Minkowski's upper bound, define limits for sphere packing densities.
  • High-dimensional sphere packing is crucial for understanding complex systems and optimizing storage/transmission.

Purpose of the Study:

  • To generate nonlattice sphere packings in high dimensions (up to 22).
  • To investigate the achievable densities of these packings.
  • To compare the generated densities against established theoretical bounds.

Main Methods:

  • Utilized the geometrical constraint satisfaction algorithm RRR.
  • Generated nonlattice packings of spheres.
  • Analyzed aggregated data from simulations in dimensions up to 22.

Main Results:

  • Achieved densities that are easily double Ball's lower bound.
  • Tentatively observed an improved exponential decay rate of density compared to Minkowski's bound of 1/2.
  • Demonstrated the efficacy of the RRR algorithm for high-dimensional packing problems.

Conclusions:

  • The RRR algorithm is effective for generating dense, nonlattice sphere packings in high dimensions.
  • The findings challenge existing density limitations and suggest new theoretical possibilities.
  • This work has implications for fields requiring efficient space-filling configurations.