Rhombic Dodecahedral Grids
<img title=”A rhombic dodecahedron” alt=”A rhombic dodecahedron” style=float:right src=https://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Rhombic_dodecahedron_4color.png/288px-Rhombic_dodecahedron_4color.png></img>
<img title=”A rhombic dodecahedral honeycomb” alt=”A rhombic dodecahedral honeycomb” style=float:right;clear:right src=https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhombic_dodecahedra.png/240px-Rhombic_dodecahedra.png></img>
Almost everyone defaults to cubes for their 3D voxels, but there are actually several shapes capable of filling 3D space. Here I’d like to explore the potential of the rhombic dodecahedron as the basis for 3D grids, as it shares some useful properties with its closest 2D equivalent, the regular hexagon:
Relationship to Spheres
Whereas a 2D hexagonal tiling is equivalent to the densest possible circle packing, the 3D rhombic dodecahedral honeycomb is similarly equivalent to the face-centered cubic (FCC) lattice, one of two densest possible sphere packings. This means it is the most accurate tiling for representing movement between neighboring cells, with the smallest average difference between taxicab and Euclidean distances.