Thursday, May 9, 2013

The stellated tetrahelix is mad-braidable 3-2-1

The stellation (omnicapping) of a triangulation is mad-weaveable, and the mad-weavers follow serpentine courses that shadow the straight courses of the weavers in a plain-weaving of the underlying triangulation. Therefore, the weaver count is in both cases the same.

The tetrahelix has a 3-2-1 plain braiding. Therefore, its stellation has a 3-2-1 mad braiding. By weaver count, this may be the smallest mad-braidable tube.

Paper net for a model of the stellated tetrahelix.

Paper model of the stellated tetrahelix with the serpentine helical paths of the mad weavers indicated. 

If the underlying triangulation is face 2-colorable (i.e., its dual is bipartite,) then the mad-woven surface has an alternate conformation in which every other stellated face is "popped inward." This alternate conformation does not alter the surface, the gross shape, or the weave in any way, but the resulting surface structure is much stiffer due to its greater corrugation.

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