Editing Magic hypercube (section)

==Magic hyperbeam==
A '''magic hyperbeam''' ('''n-dimensional magic rectangle''') is a variation on a magic hypercube where the orders along each direction may be different. As such a '''magic hyperbeam''' generalises the two dimensional '''magic rectangle''' and the three dimensional '''magic beam''', a series that mimics the series [[magic square]], [[magic cube]] and magic hypercube. This article will mimic the [[magic hypercubes]] article in close detail, and just as that article serves merely as an introduction to the topic.

===Conventions===
It is customary to denote the [[dimension]] with the letter 'n' and the [[cardinality|orders]] of a hyperbeam with the letter 'm' (appended with the subscripted number of the direction it applies to).
* '''(''n'') Dimension''' :  the amount of directions within a hyperbeam.
* '''(''m''<sub>''k''</sub>) Order''' : the amount of numbers along ''k''th monagonal ''k'' = 0, ..., ''n''&nbsp;&minus;&nbsp;1.

Further: In this article the analytical number range [0..<sub>k=0</sub>Π<sup>n-1</sup>m<sub>k</sub>-1] is being used.

===Notations===
in order to keep things in hand a special notation was developed:
* '''[ <sub>k</sub>i; k=[0..n-1]; i=[0..m<sub>k</sub>-1] ]''': positions within the hyperbeam
* '''{{angbr| <sub>k</sub>i; k=[0..n-1]; i=[0..m<sub>k</sub>-1] }}''': vectors through the hyperbeam

Note: The notation for position can also be used for the value on that position. There where it is appropriate dimension and orders can be added to it thus forming: <sup>n</sup>[<sub>k</sub>i]<sub>m<sub>0</sub>,..,m<sub>n-1</sub></sub>

===Construction===

====Basic====
Description of more general methods might be put here, I don't often create hyperbeams, so I don't know whether Knightjump or Latin Prescription work here.
Other more adhoc methods suffice on occasion I need a hyperbeam.

====Multiplication====
Amongst the various ways of compounding, the multiplication<ref>this is a hyperbeam version of (pe.): [http://mathforum.org/alejandre/magic.square/adler/product.html Alan Adler magic square multiplication]</ref> can be considered as the most basic of these methods. The '''basic multiplication''' is given by:
:<sup>n</sup>B<sub>(m..)<sub>1</sub></sub> * <sup>n</sup>B<sub>(m..)<sub>2</sub></sub> : <sup>n</sup>[<sub>k</sub>i]<sub>(m..)<sub>1</sub>(m..)<sub>2</sub></sub> = <sup>n</sup>[ [[<sub>k</sub>i \ m<sub>k2</sub>]]<sub>(m..)<sub>1</sub>k=0</sub>Π<sup>n-1</sup>m<sub>k1</sub>]<sub>(m..)<sub>2</sub></sub> + [<sub>k</sub>i % m<sub>k2</sub>]<sub>(m..)<sub>2</sub></sub>]<sub>(m..)<sub>1</sub>(m..)<sub>2</sub></sub>

(m..) abbreviates: m<sub>0</sub>,..,m<sub>n-1</sub>. (m..)<sub>1</sub>(m..)<sub>2</sub> abbreviates: m<sub>0<sub>1</sub></sub>m<sub>0<sub>2</sub></sub>,..,m<sub>n-1<sub>1</sub></sub>m<sub>n-1<sub>2</sub></sub>.

===Curiosities===

====all orders are either even or odd====
A fact that can be easily seen since the magic sums are:
:S<sub>k</sub> = m<sub>k</sub> (<sub>j=0</sub>Π<sup>n-1</sup>m<sub>j</sub> - 1) / 2

When any of the orders m<sub>k</sub> is even, the product is even and thus the only way S<sub>k</sub> turns out integer is when all m<sub>k</sub> are even. Thus suffices: all m<sub>k</sub> are either even or odd.

This is with the exception of m<sub>k</sub>=1 of course, which allows for general identities like:
* N<sub>m</sub><sup>t</sup> = N<sub>m,1</sub> * N<sub>1,m</sub>
* N<sub>m</sub> = N<sub>1,m</sub> * N<sub>m,1</sub>
Which goes beyond the scope of this introductory article

====Only one direction with order = 2====
since any number has but one complement only one of the directions can have m<sub>k</sub> = 2.

===Aspects===
A hyperbeam knows '''2<sup>n</sup>''' Aspectial variants, which are obtained by coördinate reflection ([<sub>k</sub>i] → [<sub>k</sub>(-i)]) effectively giving the Aspectial variant:
 <sup>n</sup>B<sub>(m<sub>0</sub>..m<sub>n-1</sub>)</sub><sup>~R</sup> ; R = <sub>k=0</sub>Σ<sup>n-1</sup> ((reflect(k)) ? 2<sup>k</sup> : 0) ;
Where reflect(k) true if and only if coordinate k is being reflected, only then 2<sup>k</sup> is added to R.

In case one views different orientations of the beam as equal one could view the number of aspects '''n! 2<sup>n</sup>''' just as with the [[magic hypercubes]], directions with equal orders contribute factors depending on the hyperbeam's orders. This goes beyond the scope of this article.

===Basic manipulations===
Besides more specific manipulations, the following are of more general nature
* '''^[perm(0..n-1)]''' : coördinate permutation (n == 2: transpose)
* '''_2<sup>axis</sup>[perm(0..m-1)]''' : monagonal permutation (axis ε [0..n-1])

Note: '^' and '_' are essential part of the notation and used as manipulation selectors.

====Coördinate permutation====
The exchange of coördinaat [<sub>'''k'''</sub>i] into [<sub>'''perm(k)'''</sub>i], because of n coördinates 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 "coördinaatpermutation" 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 with equal orders 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.

====normal position====
In case no restrictions are considered on the n-agonals a magic hyperbeam can be represented shown in '''"normal position"''' by:
:[<sub>k</sub>i] &lt; [<sub>k</sub>(i+1)] ; i = 0..m<sub>k</sub>-2 (by monagonal permutation)

===Qualification===
Qualifying the hyperbeam is less developed then it is on the [[magic hypercubes]] in fact only the k'th monagonal direction need to sum to:
:S<sub>k</sub> = m<sub>k</sub> (<sub>j=0</sub>Π<sup>n-1</sup>m<sub>j</sub> - 1) / 2
for all k = 0..n-1 for the hyperbeam to be qualified {'''magic'''}

When the orders are not relatively prime the n-agonal sum can be restricted to:
:S = lcm(m<sub>i</sub> ; i = 0..n-1) (<sub>j=0</sub>Π<sup>n-1</sup>m<sub>j</sub> - 1) / 2
with all orders relatively prime this reaches its maximum:
:S<sub>max</sub> = <sub>j=0</sub>Π<sup>n-1</sup>m<sub>j</sub> (<sub>j=0</sub>Π<sup>n-1</sup>m<sub>j</sub> - 1) / 2

===Special hyperbeams===
The following hyperbeams serve special purposes:

====The "normal hyperbeam"====
:<sup>n</sup>N<sub>m<sub>0</sub>,..,m<sub>n-1</sub></sub> : [<sub>k</sub>i] = <sub>k=0</sub>Σ<sup>n-1</sup> <sub>k</sub>i m<sub>k</sub><sup>k</sup>

This hyperbeam can be seen as the source of all numbers. A procedure called [http://www.magichypercubes.com/Encyclopedia/d/DynamicNumbering.html "Dynamic numbering"] makes use of the  [[isomorphism]] of every hyperbeam with this normal, changing the source, changes the hyperbeam. Basic multiplications of normal hyperbeams play a special role with the [http://www.magichypercubes.com/Encyclopedia/d/DynamicNumbering.html "Dynamic numbering"] of [[magic hypercubes]] of order <sub>k=0</sub>Π<sup>n-1</sup> m<sup>k</sup>.

====The "constant 1"====
:<sup>n</sup>1<sub>m<sub>0</sub>,..,m<sub>n-1</sub></sub> : [<sub>k</sub>i] = 1
The hyperbeam that is usually added to change the here used "analytic" number range into the "regular" number range. Other constant hyperbeams are of course multiples of this one.