Sphere packings III

This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi \mathrm{/}\sqrt{18}\approx \mathrm{0.74048...}$. This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th problem. An example of a packing achieving this density is the face-centered cubic packing. This paper carries out the third step of the program outlined in math.MG/9811073: A proof that if all of the standard regions are triangles or quadrilaterals, then the total score is less than $8\text{pt}$ (excluding the case of pentagonal prisms).