
A woven cube (a single woven voxel). 
Voxel weaving and corrugated kagome are similar weave patterns. Both are made with straight weavers having 90degree folds, in both techniques the exposed portions of each weaver are squares. Corrugated kagome always produces a corrugated surface; voxel weaving sometimes produces a corrugated surface—the exceptions are when the basket surface is parallel to the x, y, or z plane. Voxel weaving rigidly maintains angular orientation over the whole basket (i.e., each square is perpendicular to either the x, y, or z axis); corrugated kagome permits folding along the diagonal of the square (whether or not this fold is part of the corrugation pattern) so every angular orientation is possible.

A corrugated kagome unitweaving of an octahedron. The exposed portion of each weaver is a square with a diagonal fold. 
A diagonal fold in corrugated kagome corresponds to an edge in the underlying triangulation. Therefore, in flat passages with zero corrugation frequency there are not any diagonal folds. In such a passage corrugated kagome and voxel weaving are indistinguishable.

In a flat passage, zerofrequency corrugated kagome and voxel weaving are indistinguishable. The triangles are the underlying triangulation of the corrugated kagome weaving. 
Since is not possible for the triangulation of a closed shape to be entirely flat, there is always some place in a zerofrequency corrugated kagome weaving where diagonal folds appear. Therefore, there is no example a closed basket that is both voxelwoven and corrugated kagome.

Where to go from here? Since a flat passage cannot a basket make, there is no way to make a basket that is both voxelwoven and corrugated kagome. 
2 comments:
Hi James,
Loving your work. I have a question about the singlestrip triangulation approach to weave/knit/crochet any manifold surface. Do you think it is really practical, in crochet in particular, or will there be limitations on where to join stitches? I'm tempted to have a go. Is there any software that will produce a single strip triangulation?
Thanks, Tim
Tim,
I hope it's practical, but I have little practical experience. There's no need to try anything complicated for a test, ud, which is a triangular pillow, or undp, a tetrahedron, should uncover the practical issues that arise in making any topological sphere. Higher genus surfaces need to be done in topologically planar patches.
I don't have software to produce the single strip triangulation. For a simple object you could mock it up using the Flexeez construction toy. When these clever pieces are connected threeatanode each node represents a triangle. If the object is simple enough you can discover a hamiltonian circuit through the nodes by hand. (At least one such circuit exists in nearly every trivalent network, but adding boundaries—and thus some 2valent nodes—will hurt the odds. In a patch with about 15% 2valent nodes, the odds it is hamiltonian fall to 5050.) There is free graphtheory software to find all the hamiltonian circuits in a graph, but personally I find transcribing the structure into the program and running from the command line is a lot of trouble.
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