Wednesday, December 31, 2014

The difference between voxel weave and universal weave of a square grid

The difference between the voxel weave and the universal weave of a square grid—it's all a matter of where folds and cuts may be made.
The difference between the voxel weave of a square grid and its universal weave is simply a matter of where folds and cuts are permitted.

In the universal weave, the edges of the underlying graph are presumed to trace all possible locations of folds and cuts, just as they would in a mesh. This is so even though the underlying graph contains only black vertices (i.e., it locates fabric holes of only one parity.) Holes of the other parity will be interpolated by medialization, but these interpolated vertices are off-limits to folding and cutting.

In voxel weaving the same grid, the underlying graph is of course the same, but we are now also given geometric data on the white vertices (i.e., fabric holes of both parities are already located in 3D space.) In effect, we get to start with the radial of the underlying graph, and folds and cuts are permitted only along its edges which exclusively connect holes differing parity.

The 45° rotation between the two square grids of allowable folds and cuts, causes the modulus to be √2 larger in the universal weave. The fabric itself is identical.

In a plane of voxel weave, at a location where four diagonal creases can meet to form a square, a tube of square cross-section in universal weave can be woven in. This kind of join in effect permits the smaller right triangle in the knotology weaver to be used to transition between the two allowable-folds meshes.

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