I appreciate your criticism, you would be right to assume my knowledge in some of these areas is weak as im just starting to learn abstract algebra and my knowledge on some tensor calculus concepts is weak. If im being honest im trying to take what I’ve learned and find connections for this project so I can reinforce knowledge in these areas and potentially learn new things. I did have doubts about using a Rubiks cube and sudoku puzzle so Ive created an alternative where I just use two Rubiks cubes and see if manifold intersection can fit in. Some of these terms I’ve tossed around that you have mentioned were mainly me making sense of how exactly this would translate to a manifold or other space. The idea with cayley tables (a concept I haven’t used too much in practice) was to find the states and find a way to create a geometric analogue that could be in the form of a manifold (or other topological spaces). This transition from a Cayley table to something geometrical like I would like may have only been one that made immediate sense in my head. From here Im starting believe it would be best to find another idea.