More on the math of hill-and-valley weaving

Above, the quadrilateral domain of a map edge.

Above, the hill-and-valley coloring of the quadrilateral domain of a map edge via Mrs. Stott's expansion, Me(Me(m)). Black = Up; White = Down; Pink = Saddle.

Above, the hill-and-valley coloring of the quadrilateral domain of a map edge via Ring, Me(Su(m)). Black = Up; White = Down; Pink = Saddle.

From the previous post it is clear that we must start from a bipartite map (a bipartite graph embedded in a surface) in order to design a basket that can wear a hill-and-valley corrugation of its surface. There are many such maps. For example, any quadrangulation of the sphere is a bipartite map. Also, since the 3D cartesian grid of points with integer coordinates can be bicolored (e.g., simply color each vertex according to the parity of x+y+z), the surfaces of polycubes and voxelized objects (image below) are bipartite maps regardless of their (necessarily orientable) topology.

If we must start from a non-bipartite map, the map operation Radial (a.k.a., Quadrangulation), which doubles the number of edges in the map, always yields a bipartite map--the partition of vertices being respectively, the original vertices and the new vertices that are added in the center of each original face. Taking the Medial of this map (at the cost of another doubling of edges) yields a weavable pattern. So the entire chained operation on a non-bipartite map m is Me(Ra(m)). Since Medial gives the same result whether applied to a map or its dual; and Ra() and Me() of any given map are in fact duals, Me(Ra(m)) = Me(Me(m)) = Me^2(m). This compound operation, "the medial of the medial", or "medial squared", is also known as Mrs. Stott's expansion.

Another way to make a given map bipartite is to subdivide the edges, in other words, insert degree-2 vertices in the middle of each original edge. So the compound map operation needed in this case is Me(Su(M)), a map operation which I have elsewhere called Ring ("Extra ways to see: An artist's guide to map operations." Hyperseeing: Proceedings of ISAMA 2011, pages 111–121, Summer 2011. See chart below.) because it generates weave patterns associated with chainmail.

The math of hill-and-valley weaving

A weaving pattern for any graph drawn on a surface can be derived by taking the medial of the graph, a construction that fills the rhombic domain of each edge in the graph with black and white regions as below, (original edge in red, original vertices in green):

That construction suffices for ordinary weaving, but not for hill-and-valley texturing because the black regions, which correspond to the locations of the original vertices, must be partitioned into alternating hills and valleys. So we need to start with a bipartite graph (for example, see below: each edge connects a green vertex to a blue vertex), then the medial will inherit the needed bipartition of the black regions into hills and valleys from the bipartition of the original vertices.

Hill-and-valley weaving with locked crossings

Corrugated surfaces can be woven using locked crossings if the crossings are locked at some angle other than 90 degrees. The sample above was woven using crossings locked at 100 degrees (80 degrees on the acute side), with the distance between crossings fixed at 3x the width. The resulting hill-and-valley corrugation adds a lot of stiffness to the woven panel. The 1-inch wide cardstock weavers were made by hand using a paper cutter and 0.25-inch diameter hand punch. Below is the pattern used to cut the weavers.