Editing Magic hypercube (section)

==Basic manipulations==

Besides more specific manipulations, the following are of more general nature

*'''#[perm(0..n-1)]''' : component permutation
*'''^[perm(0..n-1)]''' : coordinate permutation (n == 2: transpose)
*'''_2<sup>axis</sup>[perm(0..m-1)]''' : monagonal permutation (axis ε [0..n-1])
*'''=[perm(0..m-1)]''' : digit change

Note: <nowiki>'#'</nowiki>, '^', '_' and '=' are essential part of the notation and used as manipulation selectors.

===Component permutation===
Defined as the exchange of components, thus varying the factor m<sup>'''k'''</sup> in m<sup>'''perm(k)'''</sup>, because there are n component hypercubes the permutation is over these n  components

===Coordinate permutation===
The exchange of coordinate [<sub>'''k'''</sub>i] into [<sub>'''perm(k)'''</sub>i], because of n coordinates a permutation over these n directions is required. The term '''transpose''' (usually denoted by <sup>t</sup>) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.

===Monagonal permutation===
Defined as the change of [<sub>k</sub>'''i'''] into [<sub>k</sub>'''perm(i)'''] alongside the given "axial"-direction. Equal permutation along various axes can be combined by adding the factors 2<sup>axis</sup>. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding permutation of m numbers.

Noted be that '''reflection''' is the special case:
 ~R = _R[n-1,..,0]
Further when all the axes undergo the same permutation (R = 2<sup>n</sup>-1) an '''n-agonal permutation''' is achieved, In this special case the 'R' is usually omitted so:
 _[perm(0..n-1)] = _(2<sup>n</sup>-1)[perm(0..n-1)]

===Digitchanging===
Usually being applied at component level and can be seen as given by '''[<sub>k</sub>i]''' in '''perm([<sub>k</sub>i]''') since a component is filled with radix m digits, a permutation over m numbers is an appropriate manner to denote these.