Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Hexagonal Honeycomb Conjecture

Author: Frank Morgan
Journal: Trans. Amer. Math. Soc. 351 (1999), 1753-1763
MSC (1991): Primary 52A38, 49Q20, 28A75
Published electronically: January 26, 1999
MathSciNet review: 1615934
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is conjectured that the planar hexagonal honeycomb provides the least-perimeter way to enclose and separate infinitely many regions of unit area. Various natural formulations of the question are not known to be equivalent. We prove existence for two formulations. Many questions remain open.

References [Enhancements On Off] (What's this?)

  • [A] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4, no. 165, (1976). MR 54:8420
  • [AKS] Fred Almgren, Rob Kusner, and John M. Sullivan, A comparison of the Kelvin and Weaire-Phelan foams, in preparation.
  • [CFG] Hallard T. Croft, Kenneth J. Falconer, Richard K. Guy, Unsolved Problems in Geometry, Unsolved Problems in Intuitive Mathematics, Volume II, Springer-Verlag, New York, 1991. MR 92c:52001
  • [FT] G. Fejes Tóth, An isoperimetric problem for tesselations, Studia Sci. Math. Hungarica 10 (1975), 171-173. MR 56:3748
  • [FT1] L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Die Grundlehren der Math. Wiss., Vol. 65, Springer-Verlag, Berlin, 1953. MR 15:248b
  • [FT2] L. Fejes Tóth, Regular Figures, International Series of Monographs on Pure and Applied Mathematics, Vol. 48, Macmillan, New York, 1964. MR 29:2705
  • [FT3] L. Fejes Tóth, What the bees know and what they do not know, Bull. AMS 70 (1964), 468-481. MR 29:524
  • [Fo] Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, Jason Zimba, The standard double soap bubble in $\mathbf{R}^2$ uniquely minimizes perimeter, Pacific J. Math. 159 (1993), 47-59. Featured in the 1994 AMS What's Happening in the Mathematical Sciences. MR 94b:53019
  • [GS] Branko Grünbaum and G. C. Shephard, Tilings and patterns, Freeman, New York, 1987. MR 88k:52018
  • [HHS] Joel Hass, Michael Hutchings, and Roger Schlafly, The double bubble conjecture, Elec. Res. Ann. AMS 1 (1995), 98-102. MR 97b:53014
  • [HS] Joel Hass and Roger Schlafly, Double bubbles minimize, preprint (1995).
  • [M1] Frank Morgan, The double bubble conjecture, MAA FOCUS, December, 1995.
  • [M2] Frank Morgan, Geometric measure theory: a beginner's guide, 2nd edition, Academic Press, New York, 1995. MR 96c:49001
  • [M3] Frank Morgan, Soap bubbles in $\mathbf{R}^2$ and in surfaces, Pacific. J. Math. 165 (1994), 347-361. MR 96a:58064
  • [MFG] Frank Morgan, Christopher French, and Scott Greenleaf, Wulff clusters in $\mathbf{R}^2$, J. Geom. Anal., to appear.
  • [P1] Ivars Peterson, Constructing a stingy scaffolding for foam, Science News, March 5, 1994.
  • [P2] Ivars Peterson, Toil and trouble over double bubbles, Science News, August 12, 1995.
  • [T] Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. 103 (1976), 489-539. MR 55:1208a
  • [WP] Denis Weaire and Robert Phelan, A counter-example to Kelvin's conjecture on minimal surfaces, Philos. Mag. Lett. 69 (1994), 107-110. CMP 98:05
  • [W] Hermann Weyl, Symmetry, Princeton Univ. Press, Princeton Sci. Lib. ed., 1989. MR 92b:01085

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 52A38, 49Q20, 28A75

Retrieve articles in all journals with MSC (1991): 52A38, 49Q20, 28A75

Additional Information

Frank Morgan
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267

Keywords: Hexagonal honeycomb, bees, perimeter-minimizing, isoperimetric
Received by editor(s): November 5, 1996
Published electronically: January 26, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society