It is simpler to describe the secondary polytope for the Fano plane than for the tetractys, since the Fano plane is smaller. Recall first the cover example of the green book. An associahedron edge is extended to a triangle ($2$-simplex) with the addition of chords containing the centre point of the index square. For the three dimensional associahedron, which is indexed as always by the outer hexagon of the Fano plane, there are three ways to draw point centred chords, and these new faces eminate from the three squares of the associahedron. Using a basic dimensional argument, the extra structure resembles a cube, with three source squares on the associahedron and three others reaching out to the target vertex, which is marked by the three intersecting chords.
For the tetractys point configuration, we start with the associahedron in $6$ dimensions, indexed by the $9$-gon.
7 years ago