Tuesday, June 18, 2013

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.

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