1998/11/11 by Thomas C. Hales, Hales, Thomas C.
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #math.MG
paper · pdf · doi:10.48550/arxiv.math/9811075
22 pages. Fifth in a series beginning with math.MG/9811071
arxiv created 2002/05/20 · arxiv updated 2009/11/30
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 π/√(18)≈ 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).