Monday, December 8, 2014

Basket math basics

Weaving is a matter of holes. (Image quoted from Kenneth Snelson, US Patent  6,739,937.

"Watch the donut, not the hole," is an old refrain, but it seems basket weaving works the other way. Kenneth Snelson has shown that a basket surface can be represented by its arrangement of fabric openings (a.k.a, holes) which are always to be found surrounded by openings of the opposite helical handedness.

The general method to weave a polyhedron or a map (a topological generalization of polyhedron) is to locate a hole of one helical handedness at each vertex, and a hole of the opposite helical handedness at each vertex of the dual polyhedron or map. When these two kinds of vertices are linked by edges that correspond to vertex-face adjacencies, the result is a quad-faced map called the radial of the original map (or, what is the same, the radial of its dual.) The radial is bipartite by construction, that is, its vertices fall into two classes such that no edge connects two vertices of the same class.

The radial of a map is always a bipartite, quad-faced map, the two vertex classes being the old (original) vertices and the new (inserted) vertices. (Image quoted from Extra Ways to See: An Artist's Guide to Map Operations.)

(Due to commonly invoked restrictions on what may constitute a quadrangulation, every bipartite quadrangulation is a bipartite quad-faced map, but not every bipartite quad-faced map is a bipartite quadrangulation. For example, the quads in a quadrangulation are not usually allowed to be self-adjacent, though this possibility is essential to combinatorial topology and a necessity in weaving the smallest possible baskets.)

While the radial contains all the information needed to do the weaving, its dual, the medial, is easier to read because the edges of the medial align with the weaving elements. Just as the vertices of the radial can be two-colored to reveal its bipartite nature, the faces of the medial can be two-colored in the manner of a chessboard. The same sort of chess-colored map, known as the checkerboard graph, is used in knot theory to represent alternating links.

The medial of a triangle, shown with its chess-coloring. Any triangulation (or triangle-faced map) can be woven as a basket after its triangles have been decorated in this way.

The color boundaries in the chess-coloring (here accentuated with narrow white edges) trace the paths of the weaving elements. A triangle-faced map yields what is called a kagome weave, a square-faced map yields tabby weave—and all sorts of hybrid weaves naturally crop up.

A triangulation decorated with a chess-coloring of the medial in preparation for weaving a kagome basket.

Any surface can be woven via the medial of a map. Non-orientability causes no particular difficulty for basket weaving (other than the inevitable requirement that the woven surface be able to cross through itself.)

Maps that are themselves bipartite, quad-faced maps (or are dual to such) are special because they can be woven directly without need for the refining subdivision implicit in taking the radial or medial.

An important example of bipartite quad-faced maps are the boundary surfaces of vox solids (bodies composed of unions voxels, or unit cubes) when they inherit the corners vertices and edges of the cubic network, what I will call vox surfaces.
Vox surfaces are quad-faced and bipartite. What then are they the radial of? They are the radial of a redaction of the cubic network where each cube is left represented by just four of its eight corner vertices, and the edges are a selected diagonal on each face (a total of six face diagonals) in place of the normal twelve edges of a cube. What this makes of the cubic network is an octahedron/cuboctahedron network. The octahedron/cuboctahedron network is perhaps most easily imagined as the edges in an open, point-on-point, packing of octahedra. (There is one octahedron surrounding each of the vertices redacted from the cubic network.) Carving a box solid out of this network leaves a boundary surface decorated with a single diagonal edge from each exposed cube face. Taking the radial of this embedded graph reconstitutes a bipartite and quad-faced vox surface. 

The centers of the cubic cells in a cubic network align with a second cubic network (that is to say, the cubic network is self-dual.) It is possible to pigment the cubic cells of this dual network in two colors, say red and black, such that no two cubes of the same color are face-adjacent. Carving a vox solid out of this two-colored space leaves every face of every voxel colored like four squares of a chessboard. This is exactly the chess-colored medial graph we need to weave the vox surface in the coarsest possible weave.

The somewhat finer universal weave can be carved out of two-colored space in a similar way. Each vertex in a cubic array of voxels can be enveloped by an octahedral cell in an octahedron/cuboctahedron network. Coloring each octahedral cell black and each cuboctahedral cell red, a vox solid carved out of this space will wear a chess-colored medial graph describing the universal weave of the vox surface.

A packing of octahedra and cuboctahedra. (Image quoted from

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