View attachment 4953Blah, you have inserted yourself into a discussion you have not carefully read. You also do not seem to understand the broader and deeper implications of mathematics and are spreading misinformation. Your complaints about the Rubik's cube group being discrete are silly. How can you not know that continuous groups can have discrete subgroups? Where did you see me saying that the Rubik's cube group is equivalent to SU(2)? But what makes you think SU(2) cannot have discrete subgroups? As for the comment about SU(2) versus SU(3), the first can be modeled in a space of 3 dimensions and easy to visualise as the space of the proper rotations of a 3D object. SU(3) requires a space of 8 dimensions to properly model it. Are you honestly going to insult my itelligence by claiming that it is just as easy to work with a space of 8 dimensions as one of 3 dimensions? Are you good at solving Rubik's cubes in 8 dimensions?
I like playing with 4 dimensions too, but I know how to show it to others and work out problems in Euclidean 4 Dimensial geometry that others can I see and grasp. I wonder if you can do the same? For example, one important fact about 4D geometry is that two perpendicular planes intersect in a single point in 4D space. Can you convincingly demonstrate this to high school students? This is important in the Mandelbrot Set as you can define another fractal on any point in that set that corresponds to the plane perpendicular at that point in the larger 4D space containing the Mandelbrodt Set. . These fractals are called "julia Sets".
As for S4, please enter "S4 group" into Google and see what comes up. You will see that it is shorthand for "Symmetric Group order 4" and my definition off it given above is accurate. If you guys are using it in a non-standard way, then by definition you are using jargon. You should call it something else if you specifically mean the rotational symetries of a cube, even if it is isomorphic to the conventional group S4.
Failure to trecognise the geometrical implications of your algebraic descriptions is what I am complaining about. Can you properly describe the group of rotations of a cube and its subgroups in geometric terms? this is what is so sorely needed. the most basic symmetries involving all 6 faces I call "Rotation" and "Flip". if you deal with symmetries involving 4 equitorial faces of the cube, you get "1/4 turn', and "1/2 turn" symmetries. Can you define all of these in your system of notation so that the definitions are geometrically relevant? These are not enough for the rubik's cube though. Patterns involving only edge cubies can violate parity restrictions on the rotational symmetries of the cube. Therefore, I need to add Mirror symmetries to my basic list of cube symmetries. "Reflection" is accomplished by inverting all three axes of the cube. This symmetry is shown by the classic Checkerboard pattern, you get by doing a half slice on all three middle layers of the cube. The other two Mirror symmetries are parity transforms of the Rotation and 1/4 Turn symmetries respectively. They are called "Swirl" and "Twirl". The last symmetry is 4-way and is made by applying the mirror transformation to the 6-way Flip symmetry. I call this new 4-way symmetry "Adjacent" as it creates two pairs of adjacent faces. thus we have 5 types of 6-way symmetries, 2 of which are Normal and 3 of which are Mirror. There are 3 groups of 4-way symmetries, two are Normal and one is Mirror. All 8 types of symmetries occur with the edge only patterns (such as Checkerboard) on a Normal cube. If a pattern also involves center cubies, then the Mirror patterns are only accessible on a cube that you have turned into a Mirror cube by exchanging two oppposite centers. A secondary problem is whether a pattern involves an even number or a an odd number of pair exchanges. Thus I need the Fermion orbit as well as the Boson Prime order. Most basic patterns occur in Boson/Fermion pairs. This distinction is important with the Mirror patterns too, thus I have to reckon with Mirror Boson and Mirror Fermion orbit patterns. Occasionally other orbits are involved too.
My challenge to you is that I do not believe that your purely abstract algebra approach to describing the symmetries of the cube is adequate. This is not a matter of mathematical theory, but one of linguistic description (and I have a M.A. in Linguistics too). You need to create a customized vocabulary for the Rubik's cube that expresses geometrical relationships as well as algebraic ones. And it should be straight-forward enough to be understandable to any reasonably intelligent cubist who puts forth the effort to master it. If you come up with a decent descriptive language, I will gladly abandon my idiosyncratic system for it but it must describe the geometry of cube patterns!
sincerely,
Math Bear
P.S. Blah, just in case you didnt know, SU(2) stands for "Special Unitary Group order 2". The corresponding Lie algebra is denoted "su(2)". Can you see the difference? I was using the term "algebra informally as a synonym for "group". Any group describes a basic but complete mathematical system of some sort which is what I had in mind. As I have stated before I am not trying to talk to the erudite but to ordinary, non-Sheldon-Cooper type mortals. Most assuredly, I was not thinking of Lie algebras. I am uncomfortable with their lack of asspciativity. It is easy to dispense with commutivity but I think non-associative systems are always at least a bit weird.