Key prompts to produce the code (used in Gemini 3):

• I would like you too help me write a python notebook that uses integer numpy arrays and a careful expression of square roots to express a choice of 3d coordinates of an Associahedron, so that before converting to reals, we have a formally "correct" coordinates.
• Hmmm, this actually is quite nice, however we want the raw faces, not a triangulerized form. How best to do that? We expect 6 pentagonal faces, and four quadrilateral faces.
• Hmmm, I have done enough computational geometry to know that round offs kill good algorithms. How about we go back to 4d in integers. What cn we extract in 4d, that will tells how to merge faces in 3d?
• How bout we add a few intermediate trace prints as we now have:
• Nice, we get what follows, do you see any clearn-ups we might do now?
• Ouch, this is math, we don't write one class per polytop type. Also, the next step was to generate the dual Triaugmented triangular prism and maintain the connectivity information with the elements of the associahedron, so that is another reason not to "objectify" along the proposed lines.
• Excellent, can adjust we project down to a regular 3d polytop?
• Ok, that works. Let's finish by relaxing the primal before building the dual, and when building the dual, us the relaxed coordinates to build the relaxed dual coordinates, therefore no relaxation process for the dual. This will allow the two visualizations to be sound in the duality of their coordinates.
• Nice, can you integrate the following which I added in collab to merge the two views? Also, can we build a set of master indexing vector/matrices that express all geometric relations: points to vertices, vertices to edges, etc. per polytope, and the dual bridge relations, eg faces to dual vertices, etc. To finally add yet a another plot where all of these cross relations are drawn on this last plot as small arrows, leaving from vertices, leaving from mid edges, leaving from mid face points, and directing towards their destination corresponding reverse arrow.
• Super nice. Can we use the same framework to generate a different pair of polytopes than the K5 and its dual?
• Bravo, more productive would be hard!
• Oh... you are tempting me. So can we do that? Cyclohedron or various generalized permutahedras? And then we list through all of them as multiple figures? It would be a great way to build pictures to advertise the work. Also, how would you like me to refer to your contribution when I put it on github?
• You are very good at this! Can we take those plots and put them into a single pdf?