Generalised permutohedra include the polytopes created from shrinking the square faces: so the associahedron and permutohedron in three dimensions become the $5$ and $6$ point pyramids.

The square pyramid, or octahedron, was the matroid polytope considered in MacPherson's classic paper on combinatorial differential manifolds. This is a polytope recently mentioned by Arkani-Hamed at KITP. The operad tree labels can, unsurprisingly, greatly clarify the connection to tree (and loop) diagrams for physics, since they are essentially the same thing. By removing the squares on the associahedron, we are throwing away the rooted trees with two ternary nodes, labeled by hexagons with a single chord. The permutohedron extends the associahedron by labeling trees with ordered nodes, which is the same as permuting the order of the areas between adjacent branches on the tree. Thus all matroid polytopes may be given additional labels using operads.

8 years ago

Kea,

ReplyDeleteI watched the first hour of Arkani-Hamed but did not get to the above figures, just a few minutes ago.

I am surprised no one has worked in this sort area for so many years.

The third physics of which he speaks then goes on to explain, as far as I can tell and with no exceptions is what I have shown by "Quasic Physics".

Thank you for the link for something that I can say I totally understand in pictures if not in the standard language.

The key results of residues and looping, the minors and so on, even besides recursions

is also something that can happen if we try to count all trees of things in our heads faster and faster, until these grounding ideas precipitate out. But I am well beyond the obvious intersection of just two plane vectors in the fn notation.

It was a little boring but got progressively exciting. It was a bridge to what we mean by the usual ideas of measure and physics and to our abstract methods. I liked his way of looking into the foundation of things as well of what we know already on some level. Why do we do this work anyway?

The PeSla

The PeSla, he only mentioned the octahedron very briefly. But my ears were wide open, so to speak. Why do we do this? A very good question. Beats me. After a lifetime of being discouraged by lesser mortals, if not outright abused, I would have to say it is because we cannot help ourselves. Even if it is entirely pointless, because no one listens.

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