1. Sphere Packing and Kepler’s Conjecture
Alina N. Bãduş
October 9, 2003
Mathematics Comprehensive Exam
Carleton College

2. The conjecture Kepler, 1611:
“No packing of spheres of the same radius in three dimensions has density greater than the face-centered cubic packing.”
Demonstration: how to build the fcc
It looks simple at first sight, but reveals its subtle horrors to those who try to solve it. (S. Singh)
It’s one of those problems that tells us that we are not as smart as we think we are. (D. J. Muder)

3. Definitions Sphere with center c and radius r: Sn={x | d(x,c) ≤ r}
Packing: a placing of non-overlapping spheres in space
Density δ of a packing:
the fraction of a volume filled by the spheres
δ(X,Sn) : the density of a packing of Sn with centers X
δ(Sn) = max δ(X,Sn) over all X

4. Why we care interesting problem in pure geometry; Hilbert’s list
connections with lattice theory and group theory
applications to number theory
digital communications: error-correcting codes
Easy to state, but deceivingly hard to solve!

5. Lattice Theory Point lattice Λ = a discrete subgroup
of Euclidean space (Rn)
Basis of a lattice: vectors v1,…,vm s.t. Λ={Σ kivi | ki ε }
Fundamental cell: {Σθivi | 0 ≤ θi<1}
Generator matrix of a lattice:
discriminant of Λ: det Λ = det(MMT)
We don’t care about stretching along the axis, translation or rotation (isomorphisms)
On blackboard: define the generator matrix and the discriminant
Example: the square lattice; the hexagonal (2d) lattice

6. The densest lattice packing in 2D
The hexagonal packing; Langrange & Gauss
Do the entire proof on the blackboard, using lattices

7. The densest lattice packing in 3D
Gauss; the face-centered cubic packing…
… and others!
Establish the parameters of the fcc lattice
Sketch of proof?

8. What about non-lattice packings?
Given X={x1,x2,…,xn,…},
Voronoi cell V(xi) = {y | d(y,xi) ≤ d(y,xj) for any xj ≠ xi}
→ a decomposition of space

9. The densest circle packing of all
The hexagonal packing; Thue, Fejes Tóth
Dense enough finite packing ↓
Avg. # of edges of Voronoi cell
Lower bound on vol(V(xi))
Upper bound on δ(S1)

10. Kepler’s Theorem? Maybe.
Consider a cluster of spheres and the V-cell of the central sphere.
If vol(V) is min, then δ is max.
Local optimality is not global; need correction term f.
For a cluster p, find min(vol(V(p))+f(p)).
Study the associated spherical graph
of each cluster…
6. Some graphs have automatically high
values of f; for the other 5000, use
linear optimization.

11. But is it a proof? Almost impossible to check! The use of computers
Work in progress: the Flyspeck project
Mention Hales’ “Flyspeck” project

12. Related questions Densest sphere packings in higher dimensions
The covering problem
The kissing number problem
Read again about the solutions and proofs of these problems
Add some pictures to spice up this page

13. Further research Proof through conventional methods, if possible
What about the higher dimensions?
Other convex bodies, not only spheres
Packing, covering and kissing number problems: can one configuration solve them all?
Other distances, other spaces
Define “conventional”… analogy with calculus

14. Selected References
J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer-Verlag, New York, 1998.
H. Coxeter, Introduction to Geometry, Wiley, New York, 1961.
K. Devlin, Mathematics: The Science of Patterns, Sc. Am. L., 1994.
C. Rogers, Packing and Covering, Cambridge Univ. Press, 1964.
G. Szpiro, Kepler’s Conjecture, Wiley, New York, 2003.
C. Zong, Sphere Packings, Springer-Verlag, New York, 1999.
MathWorld,
T. Hales, various papers and presentations
Mention “the bible” and talk a little bit about each source

15. ©Hugh Sitton / Tony Stone Images