What are some properties of Delone sets that come from Barlow packings of spheres? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:16:02Z http://mathoverflow.net/feeds/question/114942 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114942/what-are-some-properties-of-delone-sets-that-come-from-barlow-packings-of-spheres What are some properties of Delone sets that come from Barlow packings of spheres? mkreisel 2012-11-29T23:57:39Z 2012-12-18T19:32:42Z <p>Given a Barlow packing of \$\mathbb{R}^n\$ by balls with at most a finite number of different radii, the centers of the balls will form a Delone set in \$\mathbb{R}^n.\$ </p> <p>For a highest density sphere packing, or at least a Barlow packing of highest density among Barlow packings, must the corresponding Delone set be a Meyer set? A Patterson set? In the cases of dimensions 2 and 3 where the optimal packing using a single radius is known, the sets can be chosen to be lattices. </p> <p>I looked for papers exploring this connection and could only find this one <a href="https://www-fourier.ujf-grenoble.fr/PUBLIS/publications/REF_678.pdf" rel="nofollow">https://www-fourier.ujf-grenoble.fr/PUBLIS/publications/REF_678.pdf</a>, which is good, but doesn't address the big picture questions above. </p> <p>EDIT: As the comments indicate, we should restrict to Barlow packings. In this case the Delone sets always appear to be of finite local complexity. </p> http://mathoverflow.net/questions/114942/what-are-some-properties-of-delone-sets-that-come-from-barlow-packings-of-spheres/116732#116732 Answer by Agol for What are some properties of Delone sets that come from Barlow packings of spheres? Agol 2012-12-18T19:32:42Z 2012-12-18T19:32:42Z <p>If you consider the Barlow lattices, these are obtained by taking planar hexagonal packings of spheres, and nesting them together in layers. For each layer, there are two possible ways of placing the layer above it: <img src="http://cnx.org/content/m16927/latest/graphics15.png" alt="alt text"></p> <p>Let's normalize the Barlow lattices to have one common layer (the A-layer). The heights of the (centers of spheres of the) other layers will be multiples of the height of a tetrahedron, so are the same for each Barlow lattice. The centers of the vertices of the next layer above have two possibilities, one with projection given by the B-circles, and one given by the C-circles (these are symmetric under a transformation preserving A-circles, so let's assume the next layer is the B-circles). Then the next layer can have projection either the C-circles or the A-circles (the diagram corresponds to the face-centered cubic lattice, so the pattern there is ABCABC...). Thus, we see that for any Barlow lattice D, the differences \$D-D\$ will be a subset of the superposition of 3 face-centered cubic lattices which have the A-,B-,C-layers at each level, and thus will be a discrete set. So it will be a Meyer set (I'm going by the <a href="http://www.ams.org/mathscinet-getitem?mr=1400744" rel="nofollow">definition of a Meyer set</a> as a discrete set \$M\$ such that \$M-M\$ is also Delone). </p>