On the plane (2-dimensional space) 2-spheres
(circles) can be most efficiently packed in
the hexagonal arrangement shown above.
This naturally leads to a tiling of the
plane by hexagons whose sides are tangent
to the 2-spheres.
In 3-space (where we live) we can stack 3-spheres (oranges) most efficiently using hexagonal closest packing.
This consists of layers of spheres packed in a hexagonal arrangement, each layer fitting as snuggly as
possible into the layers below and above (a kind of "laminating" process).
In the 2-dimensional case we created tangent hexagons from the packing (tiling) of 2-spheres (circles) on the
plane. In 3-space we can in a similar way generate tangent shapes to our 3-spheres. In the animation
above a portion of one such 3-sphere packing is shown with the tangent shapes indicated (use the
buttons to rotate the thing). This shape is called a regular rhombic dodecahedron, and as the hexagon
packs the plane with no gaps, this shape - the 3-d version of the hexagon - packs 3-space leaving no gaps.
(By the way, the shape at left is also a weight diagram for the adjoint representation of the Lie group SU(4)
(no explanation provided here for this statement - SU(3) weight diagrams can be made in 2-space from hexagons).)
These two examples of packings give rise to lattices of points in 2- and 3-dimensional spaces (the centers
of the 2- and 3-spheres). These specific lattices are called laminated. The laminated lattice in
n-dimensional space is constructed from that in (n-1)-dimensional space by a layering operation similar
to that we just outlined. Without going into details (see "Sphere Packings, Lattices and Groups" by
Conway and Sloane), it turns out that perhaps the most interesting lattice of all is the laminated
lattice in 24-dimensional space. Weirder still, this is related to the fact that the only n > 1 for
which 12 + 22 + 32 + ... + n2 = k2 (a perfect square, k an integer) is n = 24 (in which case k = 70)!
This is totally mind-blowing.
If you decide to construct a regular rhombic dodecahedron (RRD), the long diagonal of each face is the
square root of 2 (1.414...) times the length of the short diagonal. You can also find them in Nature:
I have a large garnet crystal that's an RRD.