I'm not a math machine, but I dabble in dimensional stuff. I think this falls under knot theory. I have built several prototypes of a tesseract. Each of them sits in a little case in my office. One of them is made from truncated cubes, held together with elastic cord: In theory, the structure I've built is transformable through all permutations (any of its eight cubes can be rotated to the centre), thus: ...in practice, there is way too much friction. (So I've moved on to a version 4 and now a version 5.) But I learned something about this shape as I was building it. Not only does there need to be only a single elastic cord to tie it all together, but it is a continuous loop. This became obvious only after-the-fact, when I realized the vertices (all of which are identical) have an even number of edges (four) leading to them, which means there need be (and is) no start and end point. (I used to play with this puzzle: Yes, but it is not closed loop. All but two vertices have an even number of lines, so those two must form the two termini of the line.) To my question: I want to determine if it is possible - or impossible - for the path of the elastic cord to be symmetrical i.e. so that, no matter how I permute it, I cannot distinguish one orientation from another. (I don't mean rotational in 3-D space, I mean rotate it 4D space i.e. turn it inside out, so that some other cube is at the centre.) Other than diagramming at random, to find a solution, I have no way of solving it. I've tried flattening it, but that doesn't get me any closer to a solution.