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?

Summary (clusters)

Here is part 2 of the meeting summary:

Finite clusters:
  + Prove connected; enumerate combinatorics; prove unique minimum for each combinatorial type with restrictions.
 Applies to finite or periodic or on sphere.

Summary (unification)

Here is part 3 of the meeting summary:

 + Show regularity with weights?
 + Lawlor to prove double bubble in R^n and triple bubble in R^2, with feedback, before triple bubble in R^n

Summary (packings)

Here is part 4 of the meeting summary:

Disordered packings:
 + What are the most compact finite disk/sphere packings?
 + Bidisperse packings: is there one maximum in area fraction as the area ratio changes?

Summary (space-filling)

Here is part 5 of the meeting summary:

 + No space-filling foam with every F <= 13?
 + Is there a "worst" foam, in a sense to be defined?

Summary (meetings)

Here is part 6 of the meeting summary:

Forthcoming Meetings:
 + Eufoam (Lisbon, July, Vaz)
 + 20 years since Weaire-Phelan, perhaps 2014 in Cambridge (Weaire, Cox)

Tuesday, 24 January 2012

QC and WP?

Can Weaire-Phelan be beaten? Must the best space-filler be periodic? Could a quasi-crystalline structure, in the spirit of those given by Zeng et al [Nature, 248, 11 March 2004], be said to beat WP if the limiting value of its surface to volume ratio is lower than WP as the size of the quasi-crystalline region goes to infinity? The idea is attractive because of the expected preponderance of pentagons in a quasicrystal.