Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal grid (i.e., the honeycomb, illustrated above). Pappus refers to the problem in his fifth book. The conjecture was finally proven by Hales (1999, 2001).

Hales, T. C. "The Honeycomb Conjecture." 8 Jun 1999. http://arxiv.org/abs/math.MG/9906042/.

Hales, T. C. "Cannonballs and Honeycombs." Notices Amer. Math. Soc. 47, 440-449, 2000.

Hales, T. C. "The Honeycomb Conjecture." Disc. Comp. Geom. 25, 1-22, 2001.

Hales, T. C. "The Hexagonal Honeycomb Conjecture." http://www.math.pitt.edu/~thales/kepler98/honey/.

Hales, T. C. "Background on the Hexagonal Honeycomb Conjecture." http://www.math.pitt.edu/~thales/kepler98/honey/hexagonHistory.html.

Kepler, J. L'étrenne ou la neige sexangulaire. Translated from Latin by R. Halleux. Paris: J. Vrin éditions du CNRS, 1975.

Mackenzie, D. "Proving the Perfection of the Honeycomb." Science 285, 1338-1339, 1999.

Szpiro, G. "Does the Proof Stack Up?" Nature 424, 12-13, 2003. Thompson, D'A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992.

Weyl, H. Symmetry. Princeton, NJ: Princeton University Press, 1952.

CITE THIS AS:

Weisstein, Eric W. "Honeycomb Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HoneycombConjecture.html