Any partition of the plane into regions of equal area has perimeter at least that of the regularhexagonal
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." Science285, 1338-1339, 1999.Szpiro,
G. "Does the Proof Stack Up?" Nature424, 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.
