Friday, December 12, 2014

Weaving surfaces based on tetrahedral-octahedral voxels

Assembly of the tetrahedral-octahedral honeycomb. (Image quoted from TED-43, Wikipedia)

I've missed the obvious. The easiest way to obtain an all-equilateral-triangles surface approximating a given shape is to voxelize the shape in a hybrid voxel system containing tetrahedral and octahedral voxels: the tetrahedral-octahedral honeycomb.

The resulting equilateral surface can be woven directly in kagome with straight weavers:

Any equilateral surface (deltahedron) can be woven in kagome with straight weavers.
Or, each triangle can be decorated with a cube corner (either inward or outward facing) and the result can be woven in korgome (corrugated kagome) with straight weavers and nearly 100% coverage:

Any equilateral surface (deltahedron) can be woven in korgome (corrugated kagome) with straight weavers.

Images of corrugated kagome from .
The coarse weaving of the cube is identical to the korgome weaving of the tetrahedron.

Adding cube corners to the faces of a tetrahedron yields a cube. 
Cubic and TO (tetrahedral-octahedral) voxels use the same grid. We can make a TO surface by performing local surgery on a cubic vox solid, but that will not get us anywhere if a korgome weave just resurrects the coarse weave of the cubic vox solid.

A squares and equilateral triangles tiling. (Image quoted from )

Tilings of squares and equilateral triangles can easily be converted to all-equilateral-triangles by adding (or subtracting) octahedron caps at the squares.

A complex tiling of squares and equilateral triangles. (Image quoted from ) 
 When octahedron caps are side-by-side, a tetrahedron can be added between them to form a ridge.

A quasiperiodic tiling of squares and triangles. (Image quoted from Michael Baake, "A Guide to Mathematical Quasicrystals.")

The korgome weave of an octagon cap can have any combination of in and out cube corners.
Korgome weaving of the octahedral caps leaves open the possibility of whether the quadrants will be in or out, and whether the overall shape protrudes or recedes from the surface. All-out and protruding turns an isolated square into a little toadstool in a field of cube corners. (When squares are adjacent they cannot both have this toadstool form.) All-in and protruding looks like a fold-up traffic barrier. Rows of squares (these are evidently quite common in tantalum telluride quasicrystals) when filled in with tetrahedra, become korgome ridges or grooves.

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