Weaving and Petrie duality

The hemicube, Petrie dual of the tetrahedron. Sculpture by Carlo Sequin. Image quoted from  www.cs.berkeley.edu.

The universality of weaving can be expressed by stating that a basket is an embodiment of the Petrie dual of a map. The familiar sort of map duality (Poincaré duality) interchanges faces and vertices. It is a rather friendly duality because it does not take us to a different surface. But every map also has a Petrie dual. The Petrie dual interchanges faces and zig-zag paths, and, generally speaking, the Petrie dual describes a different surface. For example, the tetrahedron, which is a map of the sphere, has as its Petrie dual the hemicube, which is a map of the projective plane.

The sculpture of the hemicube by Carlo Sequin shown above exemplifies a way to visualize the Petrie dual of a spherical map. A spherical map can be realized as a wire frame with soap films stretched across its faces. To model the Petrie dual, stretch the soap films, not across faces, but a across the zig-zag circuits, that is, the sets of edges formed by alternately taking left and right turns at each vertex.

Of course, weaving elements do not follow zig-zag circuits, rather they follow central (straight ahead) circuits in the medial of the map. Deza and Dutor in "Zig-zags and central circuits for 3- or 4-valent plane graphs," show that the zigzags of a plane graph G are in one to one correspondence with central circuits of Med(G). A visual argument extends this result to non-planar trivalent maps by considering a truchet tiling of the map's triangle-faced dual, as shown below.

Both the zig-zag and the central circuit correspond to a loop of triangles in the dual—which may or may not be a Mobius strip. 

Nederland-Skoviera-Zlatos, 2001: The Petrie dual of an orientable map M is orientable if and only if M is bipartite.