tag:blogger.com,1999:blog-5446791900286151647.post4166320866176260478..comments2012-04-01T14:11:57.067-07:00Comments on Soap bubble geometry: Summary (Kelvin)Simon Coxhttp://www.blogger.com/profile/15761018190205756689noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-5446791900286151647.post-36965770368749276242012-03-27T04:58:29.423-07:002012-03-27T04:58:29.423-07:00Morgan comment continued:
Problem 17 (Phelan). We...Morgan comment continued:<br /><br />Problem 17 (Phelan). Weaire and Phelan [WP] have described a very e fficient foam, based on the A15 structure.<br />Q: Is this the optimal partition of space into equal volumes?<br />Note (Sullivan): It is provably better than Kelvin's foam [AKS].<br />Q (Sullivan): Can we prove the existence of a foam in the A15 pattern?<br /><br />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]<br />who proved that any three-manifold has a least-area fundamental domain. (See also [KS].) The following conjectures are successively stronger.<br />Conj: The Kelvin cell is the Choe cell for the BCC torus.<br />Conj: It is the least area fundamental domain for any <br />at torus of unit volume.<br />Conj: The Kelvin foam is the best partition with congruent cells.<br />Conj: It is the best partition with equal-pressure cells.<br /><br />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 ].<br />Q: Is this an optimal partition of the slab?<br />Q: Suppose we wish to divide a very long (vertical) cylinder or prism into two equal-volume halves.<br />Is the optimal way a horizontal slice halfway up?<br /><br />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.<br /><br />Problem 21 (Sullivan). A cell in an equal-pressure foam in R^3 is bounded by minimal surfaces meeting according to Plateau's rules.<br />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.<br />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.<br />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.Frank Morganhttps://www.blogger.com/profile/15570207935455948782noreply@blogger.comtag:blogger.com,1999:blog-5446791900286151647.post-4639784598274421732012-03-27T04:58:05.564-07:002012-03-27T04:58:05.564-07:00See Sullivan's Problem 18 in Section 3 appende...See Sullivan's Problem 18 in Section 3 appended below of<br />Sullivan, John M.; Morgan, Frank<br />Open problems in soap bubble geometry.<br />Internat. J. Math. 7 (1996), no. 6, 833–842.<br />MR1417788 (98a:53014)<br /><br />3. Partitions and Foams<br />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].)<br /><br />Problem 15 (Morgan). Do least-area partitions of R^n into unit volumes exist? What is the right defi nition? What regularity holds?<br /><br />[Actually existence with possibly very disconnected regions was proved in<br />Frank Morgan, Existence of least-perimeter partitions. Phil. Mag. Lett. 88 (Fortes mem. issue, Sept., 2008), 647-650; arXiv.org (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.]<br /><br />Problem 16 (for the bees). Are regular hexagons the optimal way to partition the plane into equal areas?<br />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.<br />Note (Heppes): Under the assumption of equal pressure, some optimal partitions are also known for the sphere and hyperbolic plane.<br /><br />[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.]Frank Morganhttps://www.blogger.com/profile/15570207935455948782noreply@blogger.com