On the honeycomb conjecture and the Kepler problem

TL;DR

This paper proves the honeycomb conjecture and Kepler problem by identifying optimal tiling blocks in 2D and 3D space, demonstrating that regular hexagons and specific dodecahedra are uniquely optimal for tiling and space-filling.

Contribution

It introduces a novel approach to these classical problems by framing them as extreme value problems and identifying unique minimal-volume and minimal-perimeter building blocks.

Findings

Regular hexagons are the only 2D blocks with minimal perimeter for unit area.

Rhombic dodecahedron and rhombus-isosceles trapezoidal dodecahedron are the only 3D blocks with minimal volume containing a unit sphere.

The Kepler conjecture is proved using minimal 3D Kepler building blocks.

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.