Re: [math-fun] how to make a rhombic dodecahedon
----- Original Message ----- From: Dan Asimov Sent: 06/23/13 01:18 AM To: math-fun Subject: Re: [math-fun] how to make a rhombic dodecahedon
Just to add to this, Wikipedia reminds me that there are two distinct lattice packings
Only the face-centred cubic (a b c a b c ...) is a lattice packing; the hexagonal close packing (a b a b a b ...) is not a lattice. (A lattice in R^n requires that the group of translational symmetries act transitively on the points, whereas HCP has two orbits.) Nevertheless, these are certainly the two most interesting of these 2^aleph_0 packings, by virtue of naturally occurring in crystallography. I prefer FCC due to its substantially larger point symmetry group.
that are special cases of the 2^aleph_0 packings that Adam refers to below: the cubic close-packing and the hexagonal close-packing: < http://en.wikipedia.org/wiki/Sphere_packing >. These are defined, respectively, by whether the alternation of the layers Adam mentions is in the pattern a b c a b c ... or a b a b a b ....
Also, minor nitpicking point: the enumeration of the distinct not-necessarily-lattice ways the layers can be alternated should involve the equivalence of any shifted sequence of the 3 types of layers, and also possible equivalence by rotations between two arrangements.
Good point. That doesn't affect the fact that there are uncountably many such packings, though. Sincerely, Adam P. Goucher
Thanks for setting me straight on this, Adam. Sheesh, the HCP looks so much like a lattice packing that I was fooled. --Dan P.S. This reminds me of another configuration of points in 3-space that isn't a lattice, though it's often called one informally by chemists and crystallographers: the arrangement of atoms in a diamond. This is kind of tricky to visualize fully. On 2013-06-23, at 3:11 AM, Adam P. Goucher wrote:
From: Dan Asimov
Just to add to this, Wikipedia reminds me that there are two distinct lattice packings
Only the face-centred cubic (a b c a b c ...) is a lattice packing; the hexagonal close packing (a b a b a b ...) is not a lattice. (A lattice in R^n requires that the group of translational symmetries act transitively on the points, whereas HCP has two orbits.)
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Adam P. Goucher -
Dan Asimov