This post is for kneemo, without whom it would not exist. That is, at least one person knows exactly what I am talking about!
A tetractys has an equal number of internal edges and external edges. Thus we can swap the edge weights for these two sets, obtaining a new tetractys.
This tetractys appears as a set of blue triangles in a diagram that begins with a hexagon. The hexagon has an internal triangle, just like the chorded hexagon that marks a square face on an associahedron polytope in dimension $3$. Dual to this chorded hexagon is a pink diagram, selecting $3$ pentagons and $3$ squares, namely $6$ faces of the associahedron.
6 years ago