Orbitals
Orbitals
An orbital is a group of pieces that are able to be permutated to any other piece in that orbital. The position of a certain orbital O is defined as a certain permutation of the pieces in O. The position of an orbital is solely affected by a certain set of permutations P which only affect pieces in O. (Note: the orientation of pieces in O doesn't affect the position, nor the number of orbitals).
Complete and Distinct Orbitals
A complete orbital is defined as an orbital describing every piece on a certain Rubik's cube.
A complete orbital O can be expressed as the composition of two or more sub-orbitals (O1, O2, ...)∈O if all orbitals in O exist, and if, for every sub-orbital Oi, every piece in Oi can be permuted to any other piece in Oi with a permutation affecting only pieces in Oi.
Two sub-orbitals in O, On and Om, are distinct if there are no pieces in both On and Om.
NxN Cases:
A 1x1x1 Rubik's Cube has one orbital, since there are permutations that change the arrangement of the pieces.
A 2x2x2 Rubik's Cube has only one distinct orbital, the "corner" pieces, as the corners are the only pieces on a 2x2, and are therefore the only pieces able to permutated.
A 3x3x3 Rubik's Cube has three distinct orbitals, the "corner" pieces, "edge" pieces, and "center" pieces. Notably, despite the center pieces on a standard 3x3 not being able to be reordered, there is a non-empty set of permutations that acts on the centers.
A 4x4x4 Rubik's Cube has three distinct orbitals, the "corner" pieces, the 2x2 face of "center" pieces, and the 2x1 blocks of "wing" pieces.
A 5x5x5 Rubik's Cube has six distinct orbitals, the "center", "center-edge", and "center-corner" pieces, which make up the 3x3 center face of the 5x5, the "edge wing" and "edge-center" pieces, making up the 3x1 edge block, and the "corner" pieces.
In general, an NxN cube will have (⌈N/2⌉)(⌈N/2⌉+1)/2 distinct orbials.