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| A woven cube (a single woven voxel). |
Voxel weaving and corrugated kagome are similar weave patterns. Both are made with straight weavers having 90-degree 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.
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| A corrugated kagome unit-weaving 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.
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| In a flat passage, zero-frequency 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 zero-frequency corrugated kagome weaving where diagonal folds appear. Therefore, there is no example a closed basket that is both voxel-woven and corrugated kagome.
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| Where to go from here? Since a flat passage cannot a basket make, there is no way to make a basket that is both voxel-woven and corrugated kagome. |
2 comments:
Hi James,
Loving your work. I have a question about the single-strip 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 three-at-a-node 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 2-valent nodes—will hurt the odds. In a patch with about 15% 2-valent nodes, the odds it is hamiltonian fall to 50-50.) There is free graph-theory 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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