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:


Unification:
 + 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:

Space-filling:

 + 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)

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.

Start here ...

The idea of this blog is to stimulate discussion leading up to the ICMS workshop on Isoperimetric problems, space-filling, and soap bubble geometry, Mar 19, 2012 - Mar 23, 2012

The workshop will bring together experts in geometric measure theory, numerical computation, and foam structure and applications to make progress finding and proving the optimum area-minimizing cellular structures. It is anticipated that recent computational results may provide guidance in developing proofs, and results of the last 10 years will stimulate interest and further work in the area.

Open questions include:
• What are the main difficulties in trying to use the method of proof of the honeycomb conjecture to prove that the Kelvin cell is the least area tiling of space. Can similar tools be used to say anything about the Weaire-Phelan structure and its conjectured optimality?

• What are the optimal numerical procedures for determining conjectured minimizers for finite tilings? To what extent can they aid proofs of optimality?

• Where can this mathematics make an impact?

but there are many others ...