What the Rubik's cube really is It's a device you use to "visit" a larger structure--you see one part at a time--which is composed of the faces of a "solid" with 4 dimensions. It's [a] 3-surface which initially, note that here you decide what the initial condition(s) is, has a choice of visiting one dimension of a graph made out of space and time. This 3-surface is constructed in steps, from the "axiom of choice" which is which dimension to choose to "tile:" the graph. It gets to 3x1 dimensional surfaces in spacetime and starts over. So this means that, after 3x1 surfaces have been "constructed" and used to build a 3-dimensional surface, you have part of a larger graph, and of a larger structure in that case. Abstractly, you now can use the smallest of the "full" group of permutable devices--the Pocket Cube is sliced once in each dimension--to explore the smallest subgroup of a bigger group. Mathematically, there is no reeson to not now use the 3-surface as a tile, and so get to larger "dimensions" of the range of this device. The complete map of this group as a poset (graph) is the number of "points" in each of the vertices of a triangle. The triangle looks like one tile, then like two joined along one side, then three, and all their permutations. You eventually get to a tetrrahedral subset of tiles you can use as tiles on a 4-dimensional shape. Now spacetime is 4-dimensional because of time being assigned to one of four freely chosen dimensions. I'll let you all work out the remainder of the implications of using 4-dimensional tiles, and so on. Sorry for the double posting. Anyway, one of the important things that just the smallest "configuration" of a sliced puzzle (with all of the slices intersecting at a centre) tells us is that you can only choose to not choose a direction (of rotation) four times, then you have to. If spacetime is 4-dimensional and curved, and we imagine a "hill" of it and a choice to go around this hill to the left or right, there is a third choice which is not to go around it, instead go over it which makes it look flat, not curved. The cube says you can only go over this hill four times, then you have to go around it, that is, choose a direction. You can't make the hill look flat [five] times = you can't have a hill which is 5 x "flat". In other words, this function--the cube slices (unlinks) cannot be rotated in more than four choices (of three dimensions) without choosing a direction for the fifth rotation-- is information about direction, it can't be "concealed" more than the number of dimensions in spacetime. This might explain why it gets a lot of people tied up.
Well, I think I managed to stitch together a set of conditions that correspond to "choosing a first stitch", and think of it as a thread stuck to a boundary, whose end "leaves" and passes repeatedly through a fabric using three basic kinds of stitching function. This is just a heuristic kind of way to think about what the problem is, why the boundary has a shape--what gives it this shape or restricts the "threading"? If I call X "a stitch to the left" it goes around the "hill of stuff" with the hill (a fabric) on the right. X' is called "a stitch to the right", with the fabric on the left. A repeat stitch to the left or right is an overstitch, or if this is going to be a kind of formal language, I should use "functions" like stitch_left, stitch_right, and stitch_over. So I have three ways to leave the boundary, as a thread. Left or right stitches are colored, so a stitch that looks like X,X' is a 2-color strand which is formally a function stitch_(left or right). It seems to be easier to think about what the rotations of parts of a cube are doing if you put on the kind of hat that makes the "boundary problem" look like a programming one. So I cobbled together a "language" for this boundary exploring vehicle, in order to take it through a stitching or weaving "process" along the boundary--since I know the shape, but not the "dynamics" yet. Because an overstitch can be "undone" by going over "the hill of stuff" and then "back to the left or right" there is an easy way to make this functional--a functional list of stitches, which is: [stitch_over and stitch_back(left or right)] = [stitch_(left or right) and stitch_(left or right)] = [stitch_up or unstitch]; unstitch (= stitch_undo) = [stitch_left and stitch_right] or [stitch_right and stitch_left]. Where the unstitch is a basic "data structure" that gives you something like a "normal form" of stitches, as in BNF. So that last, or basic stitch "the undo a stitch" stitch can be represented as: X'X = XX', then the equal sign represents an "or function", and composition like X -> XX is an "and function". Stitches are a meet and join graph, constructed using a simple Boolean algebra. Now if we define a "twist" function that "acts on strands" we have a way to keep track of a "stitch distance" along the boundary--any "word" the "stitching machine" can weave into this imaginary fabric will either have a "twist component" or it won't. This component is generated by stitching in the same direction over two choices of dimension. This is "in" the 2-generator subgroup.
So this idea, that there is a way to unstitch the fabric, is closely connected to solving a scrambled cube. "Scrambling" becomes a function that visits "the boundary and the interior" of this fabric. What you see and hold in your hands is a 3-dimensional object with which you choose a direction and a dimension to try to get around or over a sequence of abstract "hills" that look curved (alternatively a computer program "has" the cube--it's a data structure that it "acts" on), So you unstitch the sequence of directed and undirected edges in a graph, between nodes that are a particular one of the total "permutation space" of threads. In each case there are 3 "hills" to choose to go over, or around. The colors on the faces of the object you hold and turn or "twist" parts of, give the threads a color which is their direction, i.e. the color of a thread or strand is "left" "right" or "both"; a strand with both colors is a strand composed of "left" and its complement "right", and forms an undirected edge between nodes. An undirected node has strands connected to it which have 1 or 2 colors of direction. Some of the nodes have undirected strands that connect only "back" over the same hill, it can't "choose" an edge that increases the number of 2-color strands which are really 0-color strands--it's "stuck" in 3 dimensions at a boundary which is 4 times "both ways = twice to the left or right". Instead it has to choose one "real" hill to go around, which it does by descending into the interior of G (it chooses a backstitch in the same dimension which changes a 0-color into a 1-color strand 1-dimensionally, or a 1-color stitch in a different dimension--respectively the 1-dimensional and 2-dimensional solution to the descent problem).
