Deadmanish, your advice is helpful but a bit excessive, I prefer to develop my own style rather than follow the crowd.
Herbert, thank you for your consideration. I did check out your page. I have been involved off and on in the Rubik's cube since the original Scientific american article came out in the early 80's. It inspired me to buy a cube and it took me about 2 weeks to figure out how to solve it. I have seen that notation many times before but I prefer not to use it. I am a math teacher with many years experience and I have a very good idea of what alienates the average person with regard to mathematics and makes them give up. I am sure that sort of notation is a great convenience for those with a serious mathematical background but it is as cryptic as Elamite to even well-educated non-mathematical people. I agree that my over-reliance on verbal description is not very helpful either. I want to show examples of cubes with brief but non-technical simple geometric language to describe what I want people to see. The use of informal terms like twirl and swirl make much better mnemonics than for the average educated person than technical matheamtical notation, as long as people can clearly perceive the implied symmettry. I prefer to give directions for building patterns by telling them what basic symmetry to start from then what additional patterns and symmetry transformations to add and let them work out the final result for themselves. My scheme is designed to facilitate this as i feel it gives cubists better insight into the cube.
I might add that I am a devoted disciple of Dr. David Hestenes and his Geometric Algebra (think "real valued Clifford Algebra in a much more user-friendly format). I never truly understood Linear algebra before I studied his ideas and i have found out this is true for many others, especially those in Computer Science. He has developed a common language for theoretical physics and its associated mathematics, Dr. Hestenes innsists of geometric interprestation as essential to understanding mathematical physics and I feel it applies in many other areas as well. It is especially good for intuitive understanding. Anyway, his geometric approach inspires mine. Although you could say his language reformulates Clifford Algebra, it greatly expands its range and power and thus in a practical sense really is something new.
I think I over-emphasized SU(2). I was just thinking of it as a synonym of SO(3) and this led to confusion. In fact SO(3) is not completely adequate either as I make use of parity transfornmations (think "improper reflections") and so I should probably use O(3) as the basis for analyzing the Rubik's cube.
Rotations are interesting things. Perhaps i should say a few words about them. They are basically a 2 dimensional phenomenon and if you deal with higher dimensions you must identify them in terms of their plane of rotation which more or less describes the "direction" of rotation. Only in 3 dimensions can you have a linear "axis" of rotation that is basically defined in terms of a line perpendicular to the plane of rotation. The use of the idea of "plane of rotation" is more generally useful. It is also a good idea to think of "rotation" as something that is static or active. When static, it is a representation of an angle. If the idea of rotary motion is implied then it is active and has a definite direction, clockwise or counterclockwise. This approach also makes it easier to deal with rotations greater than 2Pi radians and is esssential in situations involving angular velocity.
I know of 5 distinct ways of representing rotations. The most basic is turns, and the easiest to visualise. One turn = 360 degrees. Then there is of course the traditional degrees which is especially good for representing angles. A lot less well known is the gradiant system developed by the French. They are apparently the only ones using it, especially in the military. This system divides the circle into 400 grads rather than 360 degrees. 100 grads is a right angle. It has certain practical convences. You can easily find the quadrant for example, or an orthogonal angle by adding or subtracting 100.
The next system is of course radians, familair to all. Because they incorporate the element of Pi in the angle measure, they simplify the form of trig functions and are especially useful for higher mathematics. They also allow for the trig functions to be defined as "trancedental" whereas this is not really true of the preceding systems since the value of a trig function of an angle that is a rational part of a full circle is necessarily algebraic (think (Cyclotomic Polynomials).
The last system is not used practically but is sometimes used for theoretical explanations. This is "Arials". These are constructed like radians but instead of calculating the arc of circumference of the unit circle the angle subtends, you instead calculate the area of the sector of the circle defined by the same arc. If you imagine a unit vector sweeping around the unit circle for one full turn, it sweeps out an area of Pi arials (=360 degrees). This is of course the same as 2Pi radians so you might consider the difference between the two trivial. The difference is conceptual. Planar area turns out to be the most natural and simple way of expressing angle measure in higher dimensions. To give an example of why arials can be conceptually useful, consider that mysterious idea that so many students have trouble accepting, namely that in the particle physics concept of spin, you need to make two full turns to get back to your starting state. With other methods of measuring rotation this looks unfathomable but not with arials. For example, do you know how long a given date, say April 1, 2015 lasts upon the Earth? The answer is 48 hours, It takes 24 hours, starting from the International Date Line for the date to sweep over the Earth and another 24 hours to be swept off of it. Likewise, if you use the arial representation of rotation , it takes one full turn to sweep out the area of the unit circle and a second full turn to erase it completely again and get back to your intial state. This example might help students to visualise the concept and accept it.
Cheers!
Math Bear
