On the plane (2dimensional space) 2spheres (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 2spheres. 

In 3space (where we live) we can stack 3spheres (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 2dimensional case we created tangent hexagons from the packing (tiling) of 2spheres (circles) on the plane. In 3space we can in a similar way generate tangent shapes to our 3spheres. In the animation above a portion of one such 3sphere 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 3d version of the hexagon  packs 3space 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 2space from hexagons).) These two examples of packings give rise to lattices of points in 2 and 3dimensional spaces (the centers of the 2 and 3spheres). These specific lattices are called laminated. The laminated lattice in ndimensional space is constructed from that in (n1)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 24dimensional space. Weirder still, this is related to the fact that the only n > 1 for which 1^{2} + 2^{2} + 3^{2} + ... + n^{2} = k^{2} (a perfect square, k an integer) is n = 24 (in which case k = 70)! This is totally mindblowing. 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. 