Isomorphisms  

In principle every regular magic hypercube is isomorphic with the set of natural numbers [1,..,m^{n}], and hence with each other. Below the more basic isomorphisms are defined which relates one hypercube to another and leave to content of the 1agonals merely reordered (1agonal invariant isomorphisms). This article is not concerned with the hypercube actual position in hyperspace, only in its orientation since it partakes in the number of aspect a hypercube has. for argument sake all coordinates range from 0 to m1, reflections introduce negative coordinates which always can be move back into the original range by adding m. (notes are added with respect to position) Also this article merely takes some notes onto the hypercubes qualifier, however as to the isomorphisms itself it is not concerned with it (the hypercube is viewed as a concept on its own) The noted "(0)position" is the one with all coordinates 0 (technically denoted as [_{j}0]) The 'axes' are all the 1agonals connected with this position. 

The position of the various language elements is important since the interchange of its application result in different hypercubes. Superscript permutations depicts transpositions, while subscripted permutation main nagonal permutation unless an ther token overides this. 

Component permutation factor: p! #[permutation] 
the interchange of hypercube components  
the interchange of the p components a hypercube has give other hypercubes p is the number of prime factors of the order m (ie: m = _{i0}∏^{p1} p_{i}) see: hypercube component squares for the intended "components". 

Component permutations combine p! hypercubes with one another through the hypercube components, the "panmagic" quality pe. is indifferent for this kind of permutation since each seperate component need be simularly qualified. 

The mentioned permutation depicts a permutation of the p components of the hypercube It thus might be usefull to define the components prior to the isomorphisms application 

Transposition factor: n! ^[permutation] 
the interchange of coordinates (axes)  
the interchange of the n axes of the hypercube can be seen as permutation when one gives the axes a number, we thus have n! aspects of the hypercube by transposition (a transposition involving only two axes can be viewed as mirroring in the 2agonal of the involved planes when viewed from above) 

With transposition the main nagonal remains fixed in its position while all the other positions change as the axes connected with the (0)position are interchanged (the hypercube in principle does not move with transposition) 

The mentioned permutation depicts a permutation of the various axes, which are numbered 0 (xaxis), 1 (yaxis) ... n1 (last axis) 

Reflection factor: 2^{n} ~R 
mirroring of the hypercube in a plane  
for an n dimensional hypercube there are n planes in all these planes there can be mirrored in an additive manner we thus have 2^{n} plane mirror images of the original hypercube 

Although the actual plane mirrored in does not really matter, it is best seen when the mirroring (of a given axis) takes place in a plane where its coordinate is 0 the mirroring is thus indicated by multiplying the axes related coordinate with 1 (in this view the entire hypercube move m places with each involved reflection, this is easily rectified by adding m to each negative coordinate) 

The mentioned reflection nuymber is a bitwise sum (ie R = _{i=0}∑^{n1} 2^{i}) for every axis i wherein the reflection takes place. 

Panrelocation Translation factor: m^{n} @[position] 
Moving hypercubes planes from one side to the other Translating the (0)position to another location (modular space viewpoint) 

This transformation moves the corner of the hypercube to any position within the hypercube or vice versa any positon in the hypercube to the hypercube corner. The lines that fall off at one end put back in on the other side of the hypercube This process thus leaves every 1agonal intact so there sums remain the same the process however breaks up the other ragonals to unbrake other ragonals In case of perfect hypercubes of course this panrelocated version is perfect still (thus the perfect hypercube is defined), in all other cases the hypercubes qualification shifts (most commenly from 'magic' to 'semimagic') in general the translation contribute a independent factor of m^{n} to the number of hypercubes reachable from a given hypercube 

From the modular space point of view this is most easely seen as a movement of the (0)position to a new location. The axes are moved along with it. In the regular view the hypercube is not considered to move at all (entire planes rotate their "identifying coordinate") 

The mentioned position is the npoint the 0position is moved to when applicated negative values of course depicts backward motion since we work on odular space. 

main nagonal Permutation factor: m!/2 _[permutation] 
permutation of the main nagonal and with it all connected 1agonals  
Permutation of the m numbers on the mainnagonal can of course be done in m! ways however since the connected 1agonals also permutes with the numbers on the mainnagonal the general permutation also garbles the numbers on the hypercubes subnagonals, the authors view is that symmetrical permutations merely permute the numbers on the subnagonals, leaving the sum invariant. Thus there are in general (modd(m))!! symmetrical permutations. The author has seen nonsymmetrical permutations being "magic property invariant" however this was due to special features the squares had. The permutation is not included in the number of aspects Since half the permutation are mirror images of the other half this contributes a dividing factor of 2, to the number of hypercube which can be obtained from the parential hypercube which parentize a family of (modd(m))!!/2 members 

In contrast to the previous three isomorphisms this one is not entirely independent of the other three, half of the permutations can also be derived from an other in conjunction with a center mirroring (reflection in all planes) 

The mentioned permutation depicts a permutation of the main nagonal, note that along with this permutation every 1agonal gets realigned with its main nagonal element. 

digit changing Permutation factor: p! (?) =[permutation] 
Changing digits in all hypercubes components  
The construction tool of changing the digits in some component of the hypercubes can be forged into a general isomorphism, attaching the digits changing permutation onto a hypercube the corresponding components are lifted out, and their digits form the indexes to the permutation. Thus each component is transformed and recombined give a new hypercube. 

The radix (factor) for the various component ought to be equal to the number of permuted digits, thus the isomorphisms are exactly defined. Corresponding factor is a bit uncertain but might be as high as p! with p the number of permuted digits 

The mentioned permutation depicts the new digits to replace the once already in the components, the component digits thus can be used as indices to the permutation 

Rotation (no factor) 
rotation of the hypercube  
There are n axes the hypercube can rotate over multiples of 90^{o} Also there are n posible rotations around the hypercubes ain nagonal 

Any rotation can be seen as a combination of an even number of reflections Also n transpositions are connected with each other by rotation, their defining permutation are connected by rotation (ie P1_{i} = P2_{(i+b)%m} for some b) Because of this the possible factor a rotation could contribute is already counted with either the more basic transposition and/or reflections. 

Related subjects  
In lite of the above discussed isomorphisms a few relations can be defined in general between hypercubes, the concepts below are not concerning the quality of the hypercube. The newly (in this article) defined concepts of Parential and GrandParential hypercubes are currently not in use but will be in future date (by this author), given the considered isomorphisms the defined "GrandParential hypercube" in the regular number range always has a '1' in its (0)position, and is considered the most basic posible hypercube, there is no considerations taken into account as to the hypercubes quality, most likely the associated qualifier is a mere "regular", higher quality hypercubes can be derived from it most likely using the below given concepts. Perfect Parential hypercubes always have a perfect GrandParential hypercube (BOLD NOT YET PROVEN HYPOTHESIS) note: Isomorphisms indicating letters are equated with their hypercube langage notation! 

Normalized positioned hypercube 
A hypercube in which the axes consist of higher (or equal) number sequences in increasing axial number. 

Parential Hypercube  A hypercube with main nagonal in ascending order in normalized position  
Least nagonal Representative LNR (LNR related hypercubes H factor; (m  odd(m))!!/2) 
A hypercube of specified quality with the smallest permutation of nagonal numbers  
LNR = P_{D} with P a parential hypercube, and _{D = _[..]} a nonsymmetric nagonal permutation 

H = LNR_{D} With _{D = _[..]} a symmetric diagonal permutation, H has same quality as LNR (broken nagonals not considered) 

GrandParential Hypercube  A parential hypercube after the lowest number is panrelocated to the (0)position  
Aspectial hypercube 
A normalized positioned hypercube transformed by a combination of transpositions and reflections, there are thus 2^{n}n! aspects 

A = N^{T}_{R} with ^{T = ^[..]} a permutation of the [0,..,n1] denoting the axial permutation _{R = ~r} a number of n bits, where each bit_{i} denote a reflection in the plane perpendicular to the i'th axis (does not really matter which plane) 

Family member hypercube 
A hypercube derived from a "Parential Hypercube" by a main nagonal permutation  
F = _{P}P with _{P = _[..]} a permutation of [0,..,n1] denoting the main nagonal permutation 

Hypercube (relationship with grandparential hypercube) 
A hypercube derived from a "GrandParential Hypercube" by a panrelocation, main nagonal permutation, transposition and reflection 

H = ^{V}_{P}G^{T}_{R} with ^{V} the panrelocation vector This combines the here defined isomorphisms and show the general way to derive any hypercube from "GrandParential hypercubes" thus combining 2^{n}n! m!/2 m^{n} hypercubes with one another. summarising this number consists of: 2^{n}: reflection _{R = ~r} n! : transposition ^{T = ^[..]} m!/2 : main nagonal permutation _{P = _[..]} m^{n} : panrelocation ^{V = @[..]} 

notes 
Parential and GrandParential hypercubes have not yet been concidered yet most likely a Parential is releted to a GrandParential by a panrelocation and a main nagonal permutation only, it might however be that also a transposition is needed to deal with its aspectial component 

most of the here counted m!/2 main nagonal permutations will disturb sub nagonals thus that the numbers won't sum to the magic sum anymore. Only (modd(m))!!/2 of the permutations merely permute the sub nagonals!! 

also the qualification of the hypercubes resulting from the panrelocation is dependent on the qualification of the starting hypercube 

yet unknown to the author: with the panrelocation in principle the main nagonal is made up of another main nagonal parallel (previously broken) nagonal. Thus with main nagonal permutation related to an other "parential hypercube" (this needs further study) of course in perfect hypercubes these parential hypercubes are related by the perfect feature (ie are panrelated) 

Hyperstar Note  
Reflection and Transposition also apply to the hyperstar, although the reflection planes are fictitious. The transposition merely changes the starpoints position on the hypersphere. (have no idea yet about numbers) 