What is the analog of the hexagonal grid in 3-dimensional space? Rhombic dodecahedral honeycomb? Conjecture: The optimal way to divide 3-space into pieces of equal volume with the least total surface area is the rhombic dodecahedral honeycomb. Reasoning: "(The rhombic dodecahedral honeycomb) is the Voronoi diagram of the face-centered cubic sphere-packing, which is the densest possible packing of equal spheres in ordinary space." (Wikipedia) This resembles very closely how the regular hexagonal grid emerges from densest circle packing. (Also in real honeycombs, where the rhombic dodecahedron appears on one side of cells.)(see wiki:Honeycomb) Furthermore the kissing number in 2-space is 6 and in 3-space it's 12. The "flat side" (edge->surface) should be where the objects meet. Therefore we expect a kind of dodecahedron. Question: Is my conjecture right? If yes, how can I proof it? References: < <

> The optimal way to divide 3-space into pieces of equal volume with the least total surface area is the rhombic dodecahedral honeycomb.
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> Is my conjecture right?

No.
It has been described in the paper "A counter-example to Kelvin's conjecture on minimal surfaces" by D. Weaire & R. Phelan (link) that both the Kelvin tetrakaidecahedron and their own structure are closer to a solution to the problem which corresponds to this question. It is called the Kelvin problem and is still an open problem in mathematics.