M Theory Lesson 69

Recall that when replacing trees by dual polygons, one can distinguish the type of the associahedron face by the kind of diagonals for the polygon. For example, the K4 Stasheff polytope has 6 pentagonal faces and 3 squares. These are distinguished by the chorded hexagons where a diagonal that splits a hexagon in two corresponds to a square. This shows how pentagons may be paired, by taking dual diagonals, but squares are at best self dual. Labelled trees may be replaced by labelled polygons.

The description of trees as clusters of polygons, used by Devadoss in tiling moduli spaces, is better known to category theorists as the theory of 2-opetopes. The dimension 2 describes the planar nature of polygons, but this may be generalised. On that note, David Corfield points out a wonderful new paper on the arxiv.

3 Responses so far »

  1. 1

    L. Riofrio said,

    Hexagoins appear truly magical. I hope that your M-Theory lessons become a book someday.

  2. 2

    Kea said,

    Thanks, Louise. Yes, it would be fun to write a book on M Theory, all based around the properties of a very simple 3d polytope.

  3. 3

    Doug said,

    Section 1.7 diagrams of the Kock, et al paper ‘Polynomial Functors and Opetopes’, referenced by Corfield, resemble electron shells about a nucleus or planetary orbits about stars except for the lack of perturbation.

    Overall the example diagrams have some resemblance to Peti Nets used in applied mathematics, but I am not sure that there is any advantage.

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