Monday, November 17, 2014

Weaving voxellated surfaces

A surface is weavable if it has a chess-colorable tessellation. The tiles of the tessellation represent the openings in the weaving, and their colors indicate the handedness (left or right) of the helix made by the weavers that go around the opening. (The weavers themselves are represented by the color boundaries between the tiles.) From that information, it is easy to deduce whether a weaver goes over or under—no matter which face of the fabric we happen to be looking at.

For example, a chessboard, as shown above, represents a simple tabby weave. A handy thing about the tabby weave is that we can pick a width for the weavers that precisely covers the surface leaving only tiny weave openings.

We can tabby-weave a cube as is shown by the coloring of the "rounded" cube in the figure above (note that the external edges of the cube, shown rounded off, are not color boundaries and thus do not represent weavers.) The pattern of the weavers forming this cube are like the three loops of string one might use to tie a parcel. The chess-coloring of the surface proves that these three loops can be put in consistent over-and-under weaving order.

Imagine that every voxel in a quantized space is like the tabby-weave cube shown above, differing in having been translated in x, y, and z to its proper position and rotated so that its corner colors match its neighbors' at that corner. Notice that every solid body formed by a union of such tabby-weave voxels is wrapped by a chess-colored tessellation. Therefore, the surface of every voxellated object can be woven in tabby weave.

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