M Theory Lesson 182

In this 1998 paper [1], Burgiel and Reiner define signed analogues of the associahedra. Recall that the vertices of an associahedron could be labelled by chorded polygons, such as the hexagon for the polytope in three dimensions. Here one uses a pentagon to obtain a three dimensional polytope. Signed squares give the octagon, as shown. Note that edges exist if either a sign or chord is flipped. There are always two vertices which remain unsigned. One wonders whether or not this particular extension is interesting in the context of operads. Does this octagon represent an octahedron in the same way that a hexagon represents a cube?

[1] New York J. Math. 4 (1998) 83-95