  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 $\pi/\sqrt{18}\approx 0.74048...$. 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 $8 \pt$ (excluding the case
of pentagonal prisms).
