Wednesday, August 24, 2011

More on ply-split braiding of baskets

Mathematically, a plain-woven basket embodies a 4-valent, chess-colorable map. Such a map can be generated from any base map M, by the map operation of taking its medial, Me(M).

A visual way to think about a map operation such as medial is as a quadrilateral truchet tiling. Every edge in a map is associated with a quadrilateral domain of the surface. The union of all of these quadrilateral edge domains is the surface itself. Ironically, the task of drawing an arbitrary map on a surface, which seems so free—a dessin d'enfant—also accomplishes the goal of sudividing the surface into quadrilaterals, a task that might seem workaday and onerous. The map operation kis, Ki(M), makes explicit the boundaries of these quadrilateral edge domains.

A quadrilateral edge domain—save for the edge itself, and a portion of the vertices at each end—is empty. Using the technique of truchet tiles we can replace this empty quadrilateral region with a tile decorated with new vertices and new edges, plus any subset of the old edge and vertices that we wish to keep. For example, a truchet tile for medial leaves out the original edge and both its vertices (for comparison, a truchet tile for the map operation identity, Id(M), reveals the original edge):

If our truchet tiling (and weaving) is not to be arbitrary, our truchet tiles must all be identical, and their placement must be automatic.

In an undirected map (i.e., one where edges are not decorated with arrows,) we discover that we have an option in placing quadrilateral tiles: we can rotate them 180 degrees in the plane, and they still line up with the old edge. In consequence, we will be adding monkey-business to our map (directing an edge) every time we place a tile—unless, of course, the decoration on the tile itself has two-fold rotational symmetry, then our choice will make no substantive difference. (Thus, amazingly, any 2D periodic structure or mechanism that can be expressed as a quadrilateral truchet tile with two-fold rotational symmetry can be applied to any map of an any surface.)

Monkey-business not being what we're looking for, we are limited to weave crossings having two-fold rotational symmetry. The ply-split crossings illustrated in the previous post lack this.

Peter Collingwood's article "Ply-Split Braiding" in Weaver's issue 29 illustrates a twined linking that does have the required symmetry:

But then we'll be back to using bent twongs.

A possibly better option, which is really just ordinary braiding with a twist, is to open out each cord into a 2x2 woven crossing:

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