In 1611, Kepler proposed that close packing (cubic or hexagonal) is the densest possible Sphere Packing (has the greatest ), and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the Kepler Problem.

A putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Hsiang 1992, Cipra 1993), but was subsequently
determined to be flawed (Conway *et al. *1994, Hales 1994). According to J. H. Conway, nobody who has read Hsiang's proof has any
doubts about its validity: it is nonsense. In 1998, Hales completed a full proof of the conjecture which appears in a series
of papers totaling more than 250 pages.

**References**

Cipra, B. ``Gaps in a Sphere Packing Proof?'' *Science* **259**, 895, 1993.

Conway, J. H.; Hales, T. C.; Muder, D. J.; and Sloane, N. J. A. ``On the Kepler Conjecture.''
*Math. Intel.* **16**, 5, Spring 1994.

Eppstein, D. ``Sphere Packing and Kissing Numbers.'' http://www.ics.uci.edu/~eppstein/junkyard/spherepack.html.

Hales, T. C. ``The Sphere Packing Problem.'' *J. Comput. Appl. Math.* **44**, 41-76, 1992.

Hales, T. C. ``Remarks on the Density of Sphere Packings in 3 Dimensions.'' *Combinatori* **13**, 181-197, 1993.

Hales, T. C. ``The Status of the Kepler Conjecture.'' *Math. Intel.* **16**, 47-58, Summer 1994.

Hales, T. C. `The Kepler Conjecture.'' http://www.math.lsa.umich.edu/~hales/countdown/.

Hsiang, W.-Y. ``On Soap Bubbles and Isoperimetric Regions in Noncompact Symmetrical Spaces. 1.''
*Tôhoku Math. J.* **44**, 151-175, 1992.

Hsiang, W.-Y. ``A Rejoinder to Hales's Article.'' *Math. Intel.* **17**, 35-42, Winter 1995.

© 1996-9

1999-05-26