Thursday, June 20, 2013

Ending up at an mmm vertex

The points-out side of a frequency-1 korgome weaving. The "points" correspond to the vertices of the primal, trivalent map.

We'll take the trivalent map as the primal description of a korgome basket because, for example, we probably have better intuition about the cube (which is trivalent) than about its dual the octahedron, and about the dodecahedron (which is trivalent) than about its dual the icosahedron.

Taking that viewpoint, the "points" in korgome weaving correspond to vertices of the primal trivalent map, and the triangles (so evident from the points-in side of frequency-1 korgome shown below) are faces of the dual triangle-faced map.

The points-in side of a frequency-1 korgome weaving. The evident triangles correspond to faces of the triangle-faced, dual map.

The path of a korgome weaver corresponds to a zig-zag (a.k.a. a left-right path or Petrie polygon) in the primal, a triangle strip in the dual, and a central circuit in the medial of either primal or dual since Me(M) = Me(M*).

The complete network of folds in frequency-0 korgome is the radial of either the primal or the dual since Ra(M) = Ra(M*). However, frequency-0 korgome is only possible if the primal map is bipartite.

(An abstract graph is bipartite if and only if its vertices can be colored "black" or "white" such that no two vertices with the same color are adjacent. If such a vertex 2-coloring of the graph exists, it is unique up to color rotation and is easily discovered by simply starting coloring.)

The complete network of folds in frequency-1 korgome is ko of the trivalent primal, Ko(M), and kis of the triangle-faced dual, Ki(M*). Frequency-1 korgome is possible for any trivalent primal map. 

Every polyphase weaver which is composed of unit weavers of even length has a w-side and an m-side (assuming, as always, we are observing the points-out face of the fabric.) Backtracking from the vertex we wish to finish weaving at, we can make that vertex mmm simply by correctly orienting its three polyphase weavers. In general, this leads to complicated constraints on the construction sequence since we must correctly orient three polyphase weavers wherever each is first encountered in the weaving.

The situation is much simpler if the trivalent primal is bipartite. Then we can start with a www vertex and sequentially add unit weavers subject to the rule that every vertex must be either www or mmm. If the primal is indeed bipartite, all will go well. At the end we will either find ourselves at an mmm vertex or right next door to one.

The cube is a simple example of a bipartite map. If we dangle a cube from one corner, we can color the bottom corner white and follow the bipartite coloring rules to color the other corners until we finish with a black corner at the top. Accordingly, in korgome weaving the basket described by this trivalent map, we should start at the bottom with a www point and make only www or mmm points until we end up at an mmm point at the top.

The cube has a bipartite map.

Being bipartite, this map admits a frequency-0 korgome weaving. In order to finish at a point-out, mmm vertex at the top corner of the cube, we can start at a point-in, www vertex at the bottom corner.

Puck weaving started with the aid of spring clamps. A completed www vertex is below, a completed mmm vertex is above. The plane can be woven with vertices of just these two types.

Another bipartite trivalent map is the (infinite) map of hexagons tessellating the plane.

The tessellation of the plane by hexagons is a bipartite, trivalent (infinite) map.

1 comment:

Rod Read said...

James, I'm going to study your blog more. Your work is impressive. I may be able to apply some of this principles to airborne wind energy kite projects. Especially relevant will be woven and stretched hexagonal meshes. My brother Murray( likes your site