Recall that internalisation turns one square into four. This occurs in the category of (stable) trees, discussed in Leinster’s book Higher Operads, Higher Categories from page 230. For example, consider the square in the diagram. This square is one of the three squares on the 9 faced Stasheff polytope, which one obtains by reducing the 5-leaved tree quadruple squares to ordinary faces, yielding a kind of classifying space. This diagram makes the contraction and expansion moves on trees more explicit. Recall that the entire polytope is labelled by a simple 1-level tree as befits a 1-operad polytope.

Observe that the total number of squares describing the Stasheff polytope is 6×5 (for the pentagons) plus 3×4 (for the squares) which comes to 42!

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## Doug said,

May 6, 2007 @ 11:22 am

Hi Kea,

John Baez TWF week 251 is interesting.

My favorite is this reference to Bob Coecke, ‘Kindergarten Quantum Mechanics’ and the discussion of “picture calculus”.

[Some pictures may worth more than a thousand words?]

http://arxiv.org/PS_cache/quant-ph/pdf/0510/0510032v1.pdf

“Picture calculus” appears to relate to your thread topic.

From my perspective these are all consistent with some type of mathematical game theory?

## Kea said,

May 6, 2007 @ 9:22 pm

Yes, the Coecke papers are great reading, and ‘picture calculus’ is exactly what we’re trying to do here.

## kneemo said,

May 6, 2007 @ 10:54 pm

Isn’t 42 also the number of vertices for the 6-factor associahedron?

## Kea said,

May 6, 2007 @ 11:30 pm

kneemo, YES, that’s the point! Louis Crane and I once had a giggle over the fact that it was

42!