What happens when I put cube and square doubles in a matrix?

[noogies]

I think I've managed to construct a symmetric set of functions, that let's a $$x, x^2, x^3\;$$ double an algebraic number.

I have a series that goes {6,9},{27,54} already. If I set x = 3, then I replace these terms with $$\; \{ 2x, x^2 \} ; \{ x^3, 2x^3 \}$$
I have the series and a symmetric set of functions; can I put these in a matrix, and what happens if I do, I'm going to need matrix functions?

This is just something that fell out of a posted list of the first two sets of combinations for the pocket Rubik's cube - is it the "doubler" I need though? I mean, there it is...

ed: whoops, bracketed as it should be.

 

[noogies]

Ok, I had a thunk about this, and the problem per se, is about composition and decomposition.

If I start with a cube, I can see 3 faces of it; This is the first d/dx = 3 square products from a cube (product). Next d/dx on 3 squares (now disjoint) gets 6x, then x. That is, 3 squares decompose into 6 edges or nodes, then into a single "X" crossing remains in G.

Take 1 x and square it; double the square, and so on to re-integrate. The 2-sliced cube is obviously algebraic, there are $$(8-1)!3^7$$ digits in the number, each is a squares product in a subgroup of squares.

But the next duple is {120,321} or {321,120} since the order of the "results" is irrelevant (each is independent functionally), and so is the operator order, except for the asymmetry: $$3x^2 \;> 2x^3$$, which implies a subtraction.

This makes sense because the outer function "looks ahead"; you can see in the chart for G that the two generators are out-of-phase at n > 0. The f o g must borrow states from q; q is the inner "algorithmic series" which is the most descent; f is least. Assume the first 15+m states for f and q are distinct, up to some value for n where borrowing means the f-list has connections to q's algebra and copies states. The f and q lists are also copies so they double |G|.

The physical cube isn't doubled its orientations are (for each crossing in X).

 

[noogies]

This is the list in question, which I believe is a density chart of the rationals in |G| = N-1.

There are 15 intervals (n) which is the same # as the values for f and q added together. The doubling of Q the quotient is seen in the two functions (full and quarter turns are "double" and "split" moves), that are out-of-phase from n = 1. These are two "path operators" because you can switch from f to q during a graph-walk. The connections between the individual quotients, and how this is ordered by n is what I'm interested in. The f gets ahead of the q function so it must borrow states from Q, or the inner quotient is generated in smaller intervals; at what point is the first congruence in f and q?

This is pursuant to reading up some papers, including Fowler's, some of Gardner's and Baez' online editorial about the symmetry groups and Rubik's especially. I realize this is only one aspect of a much larger domain.

The number f of positions that require n full twists and number q of positions that require n quarter turn twists are:
n 	f            q
0 	1            1
1 	9            6
2 	54           27
3 	321 	     120
4 	1847 	     534
5 	9992 	     2256
6 	50136 	     8969
7 	227536 	     33058
8 	870072       114149
9 	1887748      360508
10 	623800       930588
11 	2644         1350852
12                  782536
13                  90280
14                  276
.