This switching model of course speaks to solving the cube. But also to coding theory, Cayley hash tables and encryption, among other things. It represents, I think, a fairly general method of solving the problem of finding a path through a graph, by finding n paths for n inputs to n outputs, in parallel or sequentially, through a switching network fabricated from subnetworks (i.e. a switching fabric). The parallel solution speaks to parallel computation, or n algorithms acting together (so along the same timelike surface). So what you really do when you solve or try to solve, the Rubik's cube puzzle, is glue some abstract points to a boundary on which the identity acts to preserve a partial solution. So if you take a "scrambled" 3 x 3 x 3 cube and reconnect say, two vertices and an edge into a 3 x 1 row or column of "points on a boundary", you preserve this partial solution and act on the remainder with the root of the identity, on a "lower boundary". This idea is made concrete by physically gluing a vertex to something (an "upper boundary", as say a metal rod with a shaped end you can glue a corner to). Do this with the smallest puzzle, the 2 x 2 x 2. Now there are 7 permutable elements remaining, you want to glue at least one of these to the upper boundary and leave them there. That is, you quotient the space of rotations by reducing the number of elements you can rotate, until there is only one rotation left. Then you're in the first fibre, and you can close the "outputs" by leaving the puzzle one graph move from the permutation identity, a string of letters. (in the abstract, of course; physically you can claim the puzzle is solved, thus removing the identity permutation by asserting you don't have to go there).