Circle Packings and Sphere Packings

The following sections are included:

• The Problem of Circle Packings

• Sphere Packings and Crystal Formations, Kepler's Conjecture and a Least Action Principle of Crystal Formation

  • Three kinds of sphere packings and the concepts of their densities

    • Some natural problems on the density of the first kind and a generalized Dodecahedron Conjecture

  • The sphere packing problem and Kepler's Conjecture

    • A brief comparison between the sphere packing and the circle packing problem

  • Sphere packings and crystal formation

    • The least action principle and crystal formation

  • The least action principle of crystal formation and the localization of the proof of Kepler's Conjecture

• Some Basic Ideas and Crucial Understanding on which the Proofs of the Major Theorems are Based

  • Volume estimates of local cells

    • Basic geometry of local cells

    • Basic strategy of volume estimations of local cells

    • Volume estimation techniques specifically developed for local cells

    • Some examples of volume estimation

  • Geometry of single-layer local packings

    • Quantitative refinements of the problem of thirteen spheres

    • The non-tightness of local packings with twelve touching or almost touching neighbors

  • Geometry of double-layer local packing