M Theory Lesson 58

A 5-edged 2-level tree can have at most 4 input leaves. These trees define two dimensional polytopes as shown. The pentagon is familiar from 1-operad tilings, or as the Mac Lane pentagon, and the hexagon from the braiding rule for braided monoidal categories. Only polygons of six sides or less are capable of tiling the plane regularly. Here are some purely pentagonal tilings.

Higher dimensional analogues of the hexagon polytope are known as permutohedra, because they describe permutations on $(d + 1)$ letters. Loday showed that by cutting hypercubes with hyperplanes one can naturally obtain both associahedra and then permutohedra in any dimension. Postnikov et al have done a lot of work on the combinatorics of generalised permutohedra.

As a shameless thief of pretty pictures, today I offer the reader the $6_3$ knot on the Klein surface! A six punctured Klein surface has a moduli space of real dimension 24, which is a number we like a lot.

3 Responses so far »

  1. 1

    Doug said,

    Hi Kea,
    1 – The hexagon is prominent in organic chemistry.
    The pentagon is prominent in nucleic acid chemistry.

    I am fascinated by these types of ‘nature’s tilings’.

    2 – The referenced 6-punctured Klein suface is also a 3-genus quartic. The Petrie [Hamilton] cycles of figure 3 remind me of Petri Nets , although I am aware that they are not identical.

    3 – I was not aware of the work of Loday and Postnikov.
    Thanks also for these references.

  2. 2

    a quantum diaries survivor said,

    Hey Kea,

    the tilings with pentagons from your link are wonderful! I still am at a loss as to what use could be done of them, but a parquet with one of those patterns would be simply too cool…

    Totally off topic: you promised you would one day estimate the W mass, how are we doing with that ? You might be interested to know that CDF recently estimated they will reach a precision of 20 MeV with data ALREADY TAKEN. It is thus only a matter of analyzing it (which is entirely non trivial, however).

    Keep up with the great work!


  3. 3

    Kea said,

    estimate the W mass, how are we doing with that?

    Hi Tommaso! Well, Carl has some very suggestive ideas already worked out, but we are still working on an exact value. And all these funny tilings and honeycombs are a part of the problem.

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