The Magic Encyclopedia ™

Isomorphisms
(by Aale de Winkel)

This article deals with the one on one correspondences which are around in the discussed items this encyclopedia is set up for. In principle of course any hypercube is isomorphic to any other hypercube of the same order and dimension.
This article deals with 1-agonal invariant isomorphisms

Not concidering the quality of the hypercube the presented shows that 2ⁿ.n!.m!/2.mⁿ hypercubes are derivable from a single one, denoted by this author as the "Grand-Parential Hypercube".
Every seperate factor in this number are identified in this article.

Isomorphisms
In principle every regular magic hypercube is isomorphic with the set of natural numbers [1,..,mⁿ],
and hence with each other. Below the more basic isomorphisms are defined which relates one
hypercube to another and leave to content of the 1-agonals merely reordered
(1-agonal 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 m-1, 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 [_j0])
The 'axes' are all the 1-agonals 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 n-agonal 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 = _i-0∏^p-1 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 2-agonal of the involved planes when viewed from above)

With transposition the main n-agonal 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 (x-axis), 1 (y-axis) ... n-1 (last axis)

Reflection
factor: 2ⁿ

~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 2ⁿ 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∑^n-1 2ⁱ)
for every axis i wherein the reflection takes place.

Pan-relocation
Translation
factor: mⁿ

@[position] Moving hypercubes planes from one side to the other
Translating the (0)-position to another location (modular space view-point)
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 1-agonal intact so there sums remain the same
the process however breaks up the other r-agonals to unbrake other r-agonals
In case of perfect hypercubes of course this pan-relocated version is perfect
still (thus the perfect hypercube is defined), in all other cases the hypercubes
qualification shifts (most commenly from 'magic' to 'semi-magic')
in general the translation contribute a independent factor of mⁿ 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 n-point the 0-position is moved to when applicated
negative values of course depicts backward motion since we work on odular space.

main n-agonal
Permutation
factor: m!/2

_[permutation] permutation of the main n-agonal and with it all connected 1-agonals
Permutation of the m numbers on the main-n-agonal can of course be done in m! ways
however since the connected 1-agonals also permutes with the numbers on the
main-n-agonal the general permutation also garbles the numbers on the hypercubes
sub-n-agonals, the authors view is that symmetrical permutations merely permute
the numbers on the sub-n-agonals, leaving the sum invariant. Thus there are in
general (m-odd(m))!! symmetrical permutations. The author has seen non-symmetrical
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 (m-odd(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 n-agonal, note that along
with this permutation every 1-agonal gets realigned with its main n-agonal 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 n-agonal

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 Grand-Parential hypercubes are currently not
in use but will be in future date (by this author), given the considered isomorphisms the defined
"Grand-Parential 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 Grand-Parential 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 n-agonal in ascending order in normalized position

Least n-agonal Representative
LNR

(LNR related hypercubes H
factor; (m - odd(m))!!/2) A hypercube of specified quality with the smallest permutation of n-agonal numbers
LNR = P_D
with P a parential hypercube, and _{D = _[..]} a non-symmetric n-agonal permutation

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

Grand-Parential 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 2ⁿn! aspects
A = N^T_R
with ^{T = ^[..]} a permutation of the [0,..,n-1] 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 n-agonal permutation
F = _PP
with _{P = _[..]} a permutation of [0,..,n-1] denoting the main n-agonal permutation

Hypercube
(relationship with
grand-parential hypercube) A hypercube derived from a "Grand-Parential Hypercube" by a
pan-relocation, main n-agonal permutation, transposition and reflection
H = ^V_PG^T_R
with ^V the pan-relocation vector
This combines the here defined isomorphisms and show the general way
to derive any hypercube from "Grand-Parential hypercubes" thus combining
2ⁿn! m!/2 mⁿ hypercubes with one another.
summarising this number consists of:
2ⁿ: reflection _{R = ~r}
n! : transposition ^{T = ^[..]}
m!/2 : main n-agonal permutation _{P = _[..]}
mⁿ : pan-relocation ^{V = @[..]}

notes Parential and Grand-Parential hypercubes have not yet been concidered yet
most likely a Parential is releted to a Grand-Parential by a pan-relocation
and a main n-agonal 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 n-agonal permutations will disturb
sub n-agonals thus that the numbers won't sum to the magic sum anymore.
Only (m-odd(m))!!/2 of the permutations merely permute the sub n-agonals!!

also the qualification of the hypercubes resulting from the pan-relocation
is dependent on the qualification of the starting hypercube

yet unknown to the author: with the panrelocation in principle
the main n-agonal is made up of another main n-agonal parallel
(previously broken) n-agonal. Thus with main n-agonal 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 pan-related)

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)

Isomorphisms
In principle every regular magic hypercube is isomorphic with the set of natural numbers [1,..,mⁿ], and hence with each other. Below the more basic isomorphisms are defined which relates one hypercube to another and leave to content of the 1-agonals merely reordered (1-agonal 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 m-1, 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 [_j0]) The 'axes' are all the 1-agonals 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 n-agonal 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 = _i-0∏^p-1 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 2-agonal of the involved planes when viewed from above)
	With transposition the main n-agonal 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 (x-axis), 1 (y-axis) ... n-1 (last axis)
Reflection factor: 2ⁿ ~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 2ⁿ 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∑^n-1 2ⁱ) for every axis i wherein the reflection takes place.
Pan-relocation Translation factor: mⁿ @[position]	Moving hypercubes planes from one side to the other Translating the (0)-position to another location (modular space view-point)
	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 1-agonal intact so there sums remain the same the process however breaks up the other r-agonals to unbrake other r-agonals In case of perfect hypercubes of course this pan-relocated version is perfect still (thus the perfect hypercube is defined), in all other cases the hypercubes qualification shifts (most commenly from 'magic' to 'semi-magic') in general the translation contribute a independent factor of mⁿ 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 n-point the 0-position is moved to when applicated negative values of course depicts backward motion since we work on odular space.
main n-agonal Permutation factor: m!/2 _[permutation]	permutation of the main n-agonal and with it all connected 1-agonals
	Permutation of the m numbers on the main-n-agonal can of course be done in m! ways however since the connected 1-agonals also permutes with the numbers on the main-n-agonal the general permutation also garbles the numbers on the hypercubes sub-n-agonals, the authors view is that symmetrical permutations merely permute the numbers on the sub-n-agonals, leaving the sum invariant. Thus there are in general (m-odd(m))!! symmetrical permutations. The author has seen non-symmetrical 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 (m-odd(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 n-agonal, note that along with this permutation every 1-agonal gets realigned with its main n-agonal 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 n-agonal
	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 Grand-Parential hypercubes are currently not in use but will be in future date (by this author), given the considered isomorphisms the defined "Grand-Parential 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 Grand-Parential 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 n-agonal in ascending order in normalized position
Least n-agonal Representative LNR (LNR related hypercubes H factor; (m - odd(m))!!/2)	A hypercube of specified quality with the smallest permutation of n-agonal numbers
	LNR = P_D with P a parential hypercube, and _{D = _[..]} a non-symmetric n-agonal permutation
	H = LNR_D With _{D = _[..]} a symmetric diagonal permutation, H has same quality as LNR (broken n-agonals not considered)
Grand-Parential 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 2ⁿn! aspects
Aspectial hypercube	A = N^T_R with ^{T = ^[..]} a permutation of the [0,..,n-1] 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 n-agonal permutation
Family member hypercube	F = _PP with _{P = _[..]} a permutation of [0,..,n-1] denoting the main n-agonal permutation
Hypercube (relationship with grand-parential hypercube)	A hypercube derived from a "Grand-Parential Hypercube" by a pan-relocation, main n-agonal permutation, transposition and reflection
	H = ^V_PG^T_R with ^V the pan-relocation vector This combines the here defined isomorphisms and show the general way to derive any hypercube from "Grand-Parential hypercubes" thus combining 2ⁿn! m!/2 mⁿ hypercubes with one another. summarising this number consists of: 2ⁿ: reflection _{R = ~r} n! : transposition ^{T = ^[..]} m!/2 : main n-agonal permutation _{P = _[..]} mⁿ : pan-relocation ^{V = @[..]}
	notes	Parential and Grand-Parential hypercubes have not yet been concidered yet most likely a Parential is releted to a Grand-Parential by a pan-relocation and a main n-agonal 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 n-agonal permutations will disturb sub n-agonals thus that the numbers won't sum to the magic sum anymore. Only (m-odd(m))!!/2 of the permutations merely permute the sub n-agonals!!
		also the qualification of the hypercubes resulting from the pan-relocation is dependent on the qualification of the starting hypercube
		yet unknown to the author: with the panrelocation in principle the main n-agonal is made up of another main n-agonal parallel (previously broken) n-agonal. Thus with main n-agonal 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 pan-related)
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)