July 13th, 2011

beartato phd

(no subject)

A visual representation of 14, the fourth catalan number. Catalan numbers count binary trees, so there are 14 binary trees with 4 internal nodes. I drew a vertex for every such tree, and drew the trees' "skeletons", showing only the edges that connect one internal node to another, omitting leaves. If you connect pairs of trees that are related by a single tree rotation, the 14 vertices arrange themselves into a peculiar polyhedronish shape with six pentagonal faces (two of which contain the 5 binary trees with 3 internal nodes dangling to the left or right of the root) and three not-actually-planar-in-this-drawing four-sided faces. I bet you could adjust the lengths of the edges to make them planar, but I was having enough trouble in sketchup as it was, so I didn't.

---
Oh, this is in fact the Associahedron.

http://www.ams.org/samplings/feature-column/fcarc-associahedra says:

The very first description of an associahedron is to be found in the Ph. D. thesis of the topologist Jim Stasheff, but he was unable to represent it as a strict polytope. Instead, he had to be content with a somewhat curvy shape. [...] Of course, it is easy to see how to modify this to be a true polytope. In fact, John Milnor showed up at Stasheff's thesis defense with a cardboard model. But is is not at all easy to see how to straighten out Stasheff's curvy associahedron in all dimensions.