Thursday, June 20, 2013

1.5 unit weavers per vertex

Only unit weavers measuring an even number, N, squares in length are permissible if we wish to hide the splices on both of the faces. If the phase shift used in building up the composite weaver is 2 squares, the thickness of each composite weaver is N/2 plies. But, since the weavers cross two-by-two in what is locally a tabby weave, the basket itself will be N plies thick.

Each vertex in the trivalent primal map "owns" one triangle in the triangle-faced dual map. In turn, this triangle "owns" three half-squares of tabby weave. Since these three half-squares are N plies thick, each triangle (and in turn each primal vertex) "owns" 3*N/2 squares of unit weaver. Since the unit weavers themselves are N squares long, the consumption of unit weavers is 1.5 per primal vertex, regardless of length, provided the phase shift is 2.

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