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.

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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 ...