Editing Magic hypercube (section)

==Notations==

in order to keep things in hand a special notation was developed:
*<math>\left[ {}_{k} i;\ k\in\{0,\cdots,n-1\};\ i\in\{0,\cdots,m-1\}\right]</math>: positions within the hypercube
*<math>\left\langle {}_{k} i;\ k\in\{0,\cdots,n-1\};\ i\in\{0,\cdots,m-1\}\right\rangle</math>: vector through the hypercube

Note: The notation for position can also be used for the value on that position. Then, where it is appropriate, dimension and order can be added to it, thus forming: <sup>''n''</sup>[<sub>''k''</sub>''i'']<sub>''m''</sub>

As is indicated ''k'' runs through the dimensions, while the coordinate ''i'' runs through all possible values, when values ''i'' are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of ''m'', as the magic hypercube resides in n-dimensional modular space.

There can be multiple ''k'' between brackets, these cannot have the same value, though in undetermined order, which explains the equality of:

<math>\left[ {}_{1}i, {}_{k}j \right]=\left[ {}_{k}j, {}_{1}i \right]</math>

Of course given ''k'' also one value ''i'' is referred to.

When a specific coordinate value is mentioned the other values can be taken as 0, which is especially the case when the amount of 'k's are limited using pe. #''k'' = 1 as in:

<math>
\left[ {}_{k}1;\ \#k=1 \right] = \left[ {}_{k}1\ \ {}_{j}0\ ; \ \#k=1;\ \#j=n-1 \right] 
</math>
("axial"-neighbor of <math>\left[ {}_{k}0 \right]</math> )

(#j=n-1 can be left unspecified) j now runs through all the values in [0..k-1,k+1..n-1].

Further: without restrictions specified 'k' as well as 'i' run through all possible values, in combinations same letters assume same values. Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section)

Note: as far as I know this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.

Further: "'''perm(0..n-1)'''" specifies a [[permutation]] of the n numbers 0..n-1.