On the honeycomb conjecture and the Kepler problem

Fu-Gao Song; Francis Austin

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arXiv · 0906.1249

On the honeycomb conjecture and the Kepler problem

Fu-Gao Song·Francis Austin

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Abstract

This paper views the honeycomb conjecture and the Kepler problem essentially as extreme value problems and solves them by partitioning 2-space and 3-space into building blocks and determining those blocks that have the universal extreme values that one needs. More precisely, we proved two results. First, we proved that the regular hexagons are the only 2-dim blocks that have unit area and the least perimeter (or contain a unit circle and have the least area) that tile the plane. Secondly, we proved that the rhombic dodecahedron and the rhombus-isosceles trapezoidal dodecahedron are the only two 3-dim blocks that contain a unit sphere and have the least volume that can fill 3-space without either overlapping or leaving gaps. Finally, the Kepler conjecture can also be proved to be true by introducing the concept of the minimum 2-dim and 3-dim Kepler building blocks.

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Fu-Gao Song, Francis Austin. 2009-07-25. On the honeycomb conjecture and the Kepler problem. https://arxiv.org/abs/0906.1249

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