Tuesday, April 30, 2013

The smallest mad weave (anyam gila) basket

Paul Gailiunas has summarized the math and history of closed triaxial weaving, also known as anyam gila, or mad weaving. I have marked up one of his images below to emphasize the triangle grid that this kind of weaving respects.
Anyam gila weaving with its triangle grid.

Braiding background

In the usual flat braiding of three strands, the strand in the middle is exchanged with the strand on the left, then the strand in the middle is exchanged with the strand on the right, etc. Each strand thus spends only one move in the middle, but spends two consecutive moves on the right side or the left side. The weaving elements in anyam gila enjoy a similar lifestyle. They spend only one triangle inside the thickness of the fabric each time they pass through it (anyam gila is three plies thick,) they spend two triangles facing out each time they visit the front or the back face of the fabric. Overall the period is six triangles long: two in front, one in the middle, two in back, one in the middle.

The smallest anyam gila basket

The smallest anyam gila basket uses a single weaver, six triangles long, to wrap a two-sided triangle three-plies thick on each side. In the figure below, the single weaver has been cut somewhat narrow to clarify the structure.

The smallest anyam gila basket.
There are three kinds of edges, each triangle having one edge of each kind: meadow, tunnel, and scarp. The terms come from fancifully viewing the mad weave fabric as a network of wildlife overpasses and underpasses, as in this figure:

The view from one tunnel to the next: mad weaving as a network of wildlife underpasses and overpasses.

If the fabric is viewed as assembled from triangles, the meadow edge of a triangle always pairs up with the meadow edge of another triangle; scarp edges pair up with tunnel edges and vice-versa: there is a scarp above every tunnel, and a tunnel below every scarp.

Here is how the meadow/meadow fold looks in the basket above, viewed in cross-section.

Cross-sectional view of the meadow/meadow boundary in the simplest anyam gila basket.

And here is the same boundary in a larger basket where it is not folded:

Cross-sectional view of a meadow/meadow boundary in a large basket (not folded.)

At a meadow/meadow boundary nothing seems to happen---nothing changes on either side of the fabric. In the cross-sectional view we can see that a meadow/meadow boundary conceals a discontinuity in the middle stratum of the fabric. Therefore the choice of which diagonal of the quadrilateral is the meadow/meadow edge is not arbitrary, and the quadrilaterals on both faces of the fabric exactly coincide.

Here is how the tunnel/scarp fold looks in the basket above, as viewed in cross-section.
The tunnel/scarp fold in the simplest anyam gila basket. 
And here is the same boundary in a larger basket where it is not folded:

Tunnel/scarp boundary in a larger basket.

In the cross-sectional view we can see that a tunnel/scarp boundary on one side of the fabric coincides with a scarp/tunnel boundary on the other side of the fabric.

Two polarities of triangles

Joining two triangles at their meadow edges forms a quadrilateral with two scarp edges and two tunnel edges. Since the scarp edges coincide with the two parallel edges of a single weaver, the scarp edges must be opposing edges of the quadrilateral (and likewise for the tunnel edges.) Therefore, there are two possible polarities of triangles: ones that are meadow/tunnel/scarp in clockwise order and those that are the same in counterclockwise order. Each quadrilateral in an ayam gila weave is composed of a pair of triangles of the same polarity.

The polarity of the quadrilaterals tells the handedness of the weaving

The fact that there is a meadow/meadow edge dividing each quadrilateral means that there are also two possible polarities of quadrilateral: sighting down the meadow/meadow boundary we may see the scarp on the left or the right. The quadrilaterals in an anyam gila weave are all of the same polarity, including those on the reverse face. (The meadow/meadow boundary is in the same location on either face; going to the other side reverses our orientation, but the tunnel/scarp to scarp/tunnel correspondence compensates for the reversal.) The polarity of any quadrilateral tells the handedness of the weaving.

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