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.
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I do not know, but I see no reason why this couldn't be non-periodic. A while a go we looked at the dual of the 3D Penrose tiling. Unfortunately the structure contains also vertices of valence greater than 4.
ReplyDeletehttp://science.unitn.it/~gabbrielli/javaview/pt2.html
Regarding the article by Zeng et al the correct reference is Nature, 428, 11 March 2004. The periodic structure described there is the sigma phase found by Frank and Kasper in 1958, which does not beat Weaire-Phelan. The quasi-crystalline structure is also unlikely to improve upon the current minimizer, as it is build out of A15 and Z units too. It is worth to note that all these structures miss 13-sided polyhedra.
ReplyDeleteA first step to proving Kelvin the optimal single tile would be to do so after modding out by some of the Kelvin symmetries, first of all in the space obtained by identifying faces of a single Kelvin tile.
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