Monday, 26 March 2012

Summary (Kelvin)

Here is part 1 of the meeting summary:

Kelvin problem:
  All unit volume, all have polyhedral (and perhaps Voronoi) analogs; all have local analogs.
  + Kelvin best tiling with its combinatorics and all its symmetries?
  + Kelvin best tiling by BCC translations?
  + Kelvin best tiling by translations?
  + Kelvin best isohedral tiling?
 (Kelvin best monohedral tiling?)
 (Kelvin best equal-pressure?)
  + Kelvin best for Dehn invariant 0 (Hales)?
  + Weaire-Phelan best tiling with its combinatorics and all of its symmetries?
  + Weaire-Phelan best dihedral tiling?
  + Weaire-Phelan best tiling?
  + Kelvin best Voronoi for any lattice (Hales)? Needs bound on area ...
Sullivan: S >= 2/width. Hence if w =< 3/8, mu = (S/2)^3/V^2 ~ 18.963.  (WP ~ 18.48).
Lattice must be critical; maybe there are just five, and only Kelvin stable.
  + Can any of these be solved more easily in R^4?


  1. See Sullivan's Problem 18 in Section 3 appended below of
    Sullivan, John M.; Morgan, Frank
    Open problems in soap bubble geometry.
    Internat. J. Math. 7 (1996), no. 6, 833–842.
    MR1417788 (98a:53014)

    3. Partitions and Foams
    Kelvin's problem asks for the optimal partition of space into equal-volume cells. The solution is expected to have the geometry of a foam, but not even existence is known for such infi nite clusters. (See [Bro, Prob. 1.2].)

    Problem 15 (Morgan). Do least-area partitions of R^n into unit volumes exist? What is the right defi nition? What regularity holds?

    [Actually existence with possibly very disconnected regions was proved in
    Frank Morgan, Existence of least-perimeter partitions. Phil. Mag. Lett. 88 (Fortes mem. issue, Sept., 2008), 647-650; (2007). On regularity, see Sect. 11 of Brian White, Existence of least-energy configurations of immiscible fluids, J. Geom. Anal. 6 (1996), 151-161. These references and more appear in the 2009 edition of Morgan's Geometric Measure Theory book.]

    Problem 16 (for the bees). Are regular hexagons the optimal way to partition the plane into equal areas?
    Note (Sullivan): Results of Fejes T oth [FT1, xIII.9] [FT2, x26] together with a truncation argument [Mor6] imply this is best if the cells have equal pressure.
    Note (Heppes): Under the assumption of equal pressure, some optimal partitions are also known for the sphere and hyperbolic plane.

    [For plane proved by Thomas C. Hales, The honeycomb conjecture, Disc Comp. Geom. 25 (2001), 1-22. See Morgan's Geometric Measure Theory, Chapt. 15. See also Quinn Maurmann, Max Engelstein, Anthony Marcuccio, and Taryn Pritchard, Asymptotics of perimeter-minimizing partitions, Canadian Math. Bull. 53 (2010), 516-525.]

  2. Morgan comment continued:

    Problem 17 (Phelan). Weaire and Phelan [WP] have described a very e fficient foam, based on the A15 structure.
    Q: Is this the optimal partition of space into equal volumes?
    Note (Sullivan): It is provably better than Kelvin's foam [AKS].
    Q (Sullivan): Can we prove the existence of a foam in the A15 pattern?

    Problem 18 (Sullivan). Kelvin's foam, based on the BCC lattice, is not the optimal partition of space into equal volumes, though it is the best with its combinatorics and symmetry [AKS] [never published]. It might be optimal for various other restricted problems, like that considered by Choe [Cho1]
    who proved that any three-manifold has a least-area fundamental domain. (See also [KS].) The following conjectures are successively stronger.
    Conj: The Kelvin cell is the Choe cell for the BCC torus.
    Conj: It is the least area fundamental domain for any
    at torus of unit volume.
    Conj: The Kelvin foam is the best partition with congruent cells.
    Conj: It is the best partition with equal-pressure cells.

    Problem 19 (Heppes). Suppose we have an optimal cluster or partition of a region in the plane,and extend it to a partition of a slab in space by crossing with a short interval [0, epsilon ].
    Q: Is this an optimal partition of the slab?
    Q: Suppose we wish to divide a very long (vertical) cylinder or prism into two equal-volume halves.
    Is the optimal way a horizontal slice halfway up?

    Problem 20 (Kraynik). The regular hexagonal foam in the plane has equal shear modulus in all directions. Weaire, Fu and Kermode [WFK] have conjectured that no two-dimensional froth (with average bubble area one) has greater shear modulus in any direction.

    Problem 21 (Sullivan). A cell in an equal-pressure foam in R^3 is bounded by minimal surfaces meeting according to Plateau's rules.
    Conj: There are only finitely many possible combinatorial types for such cells. In particular, it seems unlikely that tetrahedra can occur. Probably not even a dodecahedron can occur, which would imply that none of the foams based on TCP structures can be made equal pressure.
    Note: Several types of cells are known, including Kelvin's truncated octahedron, and all have at least 14 faces. Kusner has shown [Kus] that the cells in an equal-pressure foam must have at least 13:39 sides on average.
    Note: A dihedral cell could exist only if there is a boundary curve in R^3 which bounds two minimal surfaces meeting at 120 angles along its entire length.