Sphere Packing with Rhombic Dodecahedra

(Email of November 9, 1997 from Dr. Donald Carlson).

Mark Newbold,

Thank you for your excellent Web site! The graphics in your "Recipe for a Rhombic Dodecahedron" were just what I was looking for to help me write the equations of the RD's twelve planes. As a nuclear criticality safety engineer, I will be using these equations in my computational studies of fission chain reactions in close-packed arrays of identical spheres. After searching many other Web sites, and more than a few text books in math and material science, I have been surprised not to find any mention of the rhombic dodecahedron's relevance to sphere close-packing. If you know of any such references, please let me know. Meanwhile, a fairly detailed description of my quest to use RD's to model sphere packings may be in order:

An infinite lattice of spheres can be modeled as a single sphere inscribed in a unit cell with reflecting boundaries (i.e., a polyhedral box of mirrors). For a 2D close-packing, each sphere is in contact with six neighbors, so it is easy to see that the unit cell consists of a sphere contained in a mirrored right hexagonal prism. In a 3D close-packed lattice, each sphere has an additional three neighbors on top and three on bottom, giving a total of twelve neighbors. There are two kinds of 3D close-packings to consider: 1) the face-centered-cubic (fcc) lattice, and 2) the hexagonal-close-packed (hcp) lattice. In the fcc lattice, the top three and bottom three spheres are offset by 180 degrees (or 30 degrees), while they are directly opposite each other in the hcp lattice. For both cases, the unit cells can be described as a right hexagonal prism cut by three slanting planes on the top and three slanting planes on the bottom.

In June of this year, I finally discovered in George Hart's Web pages (http://www.georgehart.com/virtual-polyhedra/dodecahedra.html) the fcc unit-cell container I had envisioned and learned its name: the normal rhombic dodecahedron. I also found there the twisted rhombic dodecahedron, corresponding to the hcp cell, as well as an elongated dodecahedron, which one could use for close-packings of rounded capsules or oblong ellipsoids (i.e., eggs). (Actually, the twisted rhombic dodecahedron is a bit more complicated in that its hcp unit cell calls for a combination of mirror-reflective and periodically-reflective surfaces; more on that later). I immediately wrote an e-mail message about my work to Dr. Hart, to which he responded by putting me in touch with Pan Dragon (http://www.pandragon.com/polyhedra/geometry.shtml). Both Hart and Pan Dragon seemed unaware of the RD's connection with sphere-packing until I brought it to their attention. At my suggestion, Pan Dragon soon created and posted some rhombic-dodecahedral sphere-packings (23JUN97) and oblong-ellipsoid-packings, or egg-packings, (14JUL97) in VRML, with animations. Actually, Pan Dragon's egg-packing has its elongated ellipsoidal axis in the stacking plane, rather than perpendicular to it (i.e., see Hart's elongated RD), and is therefore probably not a true close-packing. I also suggested in one of my e-mails to Pan Dragon that teachers and scientists might enjoy playing with transparent rhombic-dodecahedral (both normal and twisted) building blocks with colored or tinted spheres at their centers. Perhaps someone has already created such building blocks, but I haven't found them advertized anywhere.

Your RD "recipe" graphics make it easy to visualize what is needed for both the fcc and hcp cases. Using Hart's twisted rhombic dodecahedron (TRD) for the infinite hcp case is complicated by a unit-cell symmetry that calls for a combination of mirror and periodic reflecting surfaces. To get around this, one can simply slice off the "bottom" half of the normal rhombic dodecahedron (NRD) and its inscribed sphere with a mirror-reflecting plane that is normal to and bisects one of the cube's diagonals. This reflecting plane coincides with the equatorial plane of the 2D hexagonal packing layer. The desired hcp unit cell then consists of the remaining "top" half of the sphere and NRD, with mirror-reflective surfaces on all nine sides. Finite hcp stackings, on the other hand, may actually call for construction of a full TRD, but, of course, without the need for reflecting planes. This can be done by rotating the three bottom planes of the full NRD 60 degrees about the sphere axis normal to the 2D equatorial plane.

Again, I would appreciate it if you could direct me to any Web sites or literature that discuss RD's in relation to sphere packings. Please feel free to forward my messages as appropriate. Thanks again for your wonderful pages.

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