Thursday, December 11, 2014

Weaving non-cubic voxels

Down-sampling the cubic grid yields two alternative, non-cubic shapes for voxels: the rhombic dodecahedron (down-sampled 2:1) and the truncated octahedron (down-sampled 4:1.)
Cubic voxels are easy, but they have problems. From a structural point of view, their big problem is anisotropy: depending on the orientation of the cubic grid, cubic voxels may alternately produce surfaces that are smooth (and therefore un-corrugated and flexible) and or quite rough (and therefore highly corrugated and stiff.) Down-sampling the cubic grid in the two most symmetrical ways (FCC and BCC) leads to voxels in two new shapes: the rhombic dodecahedron and the truncated octahedron. These non-cubic voxels are more isotropic than cubes.

Weaving of a surface based on voxels that are truncated-octahedral is complicated by the fact that the faces are not congruent: there are both square and hexagonal faces.

This weaver is both crooked and of varying width—and yet it fails to fully cover the hexagonal faces of the truncated octahedron.

Weaving surfaces that are faceted with truncated octahedra is complicated by the fact that the faces are not congruent, there are both squares and hexagons. The worst consequence of this is that the order of faces a weaving element crosses may change as voxels are added to or subtracted from the vox solid.

Weaving of a rhombic dodecahedron is simplified by the fact that the faces are all the same shape.

Weavers for the rhombic dodecahedron are crooked but they have constant width.
Weaving is simpler when the voxels are rhombic dodecahedra. All the faces of a rhomic dodecahedron are the same shape, and, though the weavers are crooked, their alternation "left-diamond, right-diamond" does not change as voxels are added or subtracted from the vox solid.

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