Color coding and compression of codes:

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Consider that the elements of a Rubik's cube are composed of black plastic. The assembled structure is then [what I will bang ahead and write as] a B-structure, or a B, with structure (i.e. black plastic with structure). There is a set g which maps B to itself. This structuring uses black as an abstract basis and plastic as a physical basis, hence there is a space within a space.

The structure set S is open, since structure is based on d a number of spatial dimensions, and N a number of sections of B which form a cell (of color elements). B is also the base for a number of colors (C), also an open set.

The group of operations on B is closed under the structure group of S in SO(n=d), S includes the slice group of G which is the set of planes in position, pairwise parallel or orthogonal (i.e. that form a cube-relative shape or cuboid).

Together C and B form a code-group where b is the number of 0 bits in any code, that can be written or rewritten in C.

C is degenerate, and is a function of arity of the b-sets. In B, this correlates to a mapping function F.

F is the set containing the spatial encoding (the face group) which is closed under rotation, and association (since the faces associate cube-wise, by construction and therefore by any map to B).

The rotation generators, or rotors, induce transpositions in the code word W, which is the "inital map" at T.

Since C is essentially a set of pairwise subsets of color, C has 2-arity by construction. There is a triadic component in G the generator set, which is divided by C, over b the number and order of zeros in W. This component is the set K of 3-faceted elements of B, when they have a pair of colors c in C, and C is closed. The pair can be on two separate elements of K or on the same element[, which is the swapping function].

The set of colors is arbitrarily chosen from a universal set, and C also corresponds to a choice of spacetime [frequency and periodicity] coordinates, for G. This is the generally relative nature of a set of closed operations in G (the group of symmetries of space and time), and of the encoding basis (anything written "in G"). That's general coding, relative to B.

Relations are between basis changes, over F which preserve the structure and the information content. Since rotations preserve the structure of B, then if B has a chosen basis the rotors are 'structure and content'-preserving operators.

Codes W are encoded and decoded (in T) cyclically, any subset of rotations encoded as a word, which represents an instruction for an abstract machine, will have a finite number of iterations in G, recursively. Iterations are steps Tn in the cyclic transposition of the code word written in the basis b,c, where c is a 2x2-ary code (i.e. pairwise complemented sets of the b-sets). This 2s-complement encoding of W, is the quotient function [space] Q|F.

A version of the Pocket Cube exists, dubbed the "Junior Cube". This version has only two colors."

--www.noshit.org

So the Junior version multiplies the b-sets by 2. This reduces the arity of the "c-sets". C in this sense is both color and complements of color, so there is a map from C to itself. The map from C to B is open, it represents the "color-channel" for F. B is a carrier for the C-code sets. The twist group of generators introduces parity to the 3-facet subgroup. Groups of elements with 2 facets (the 2-facet group of the structure set) have one less degree of parity than K, which is the Pocket Cube's vertex set. "Twist" is the parity generator.

B has the character "the absence of color", a function W is assumed which has the character "the presence of color", and both B and W are true for all colors C and functions of color F.

The term "arity" is primarily used with reference to operations. If f is the function f : S^n → S, where S is some set, then f is an operation and n is its arity.

[n is the logarithm of f(S), mathematically.]

Arities greater than 2 [ = log2] are seldom encountered in mathematics, except in specialized areas, and arities greater than 3 [ = log3] are seldom encountered in theoretical computer science. Practical computer programming commonly defines functions with many arguments, say N, but there is a matter of use of conventions (in the particular language as well as in the culture of the programmers), whether these are regarded as N separate arguments or as an N-tuple of (often heterogenously) typed arguments.

In mathematics, depending on the branch, arity may be called type, adicity or rank.

In computer science arity may be called adicity, a function that takes a variable number of arguments being called variadic.

In linguistics and in logic, arity is sometimes called valency, not to be confused with valency in graph theory.

However, the facets of B, including C are logarithmic ratios, as the arity of functions F3/F2. |G| is a graph, with T edges and nodes, so T is graph-symmetric.

The logarithm of B "to the C" is C, n is the arity of B. The logarithm of G is n, this is known to be 11 for K. If S (the structure group) is a superset of K it has 11-arity in G when N = 2. So we assume any M or abstract machine we can use, which has a structure and a color basis included in W, and word it can read or write is at least 2-dimensional, or the T is a tape of some kind with symbols S on it.

So the S1, S2, S3 above are abstract logarithmic structures of some kind we want to decode.

We abstract this notion of decoding W or any word "in W which is in the basis b,c" to having first decoded K's arity and adicity. That was Rubik 101, chapter 1, we are still decoding chapter 2, another logarithmic function of C. C in this sense is computation(al), and we have log2 machines M, with flat "screens" log2(S).