The Magic Encyclopedia ™

The Hypercube Construction
(by Aale de Winkel)

The Construction of hypercubes is one of the major points of our investigation, this article summarizes the methods this author knows of and serves as a reference for future programs.
The Construction of a hypercube is also called a "production" (in hypercube language terminology)
NOTE: currently this article must be seen as preliminairy, some issues might change slightly when implemented, in case I'll intend to keep this article up to speed.
NOTE: this article is not too concerned with hypercube quality, the "productions" itself are most general, finetuning is needed to obtain hypercubes of certain quality.

The Hypercube Construction
Most values for parametrized construction methods will produce hypercubes of low quality

Knight Jump Vectors Starting from the position of 1 a series of n vectors fill the hypercube
The position of the '1' in the hypercube is given by a n-Point within the hypercube
the step toward the number '2' is given by a n-Vector with entries relative prime to the
hypercubes order, after n such steps an occupied place is reached and a second n-Vector is
needed too reach a non-occupied place to continue stepping with the first n-Vector thus a
total of n n-Vectors are needed to fill up an n-dimensional hypercube.
notation n by n+1 matrix (KJ)
first column the n-Point (position of '1')
2nd column the n-Vector from '1' to '2'
3rd column the n-Vector from 'm' to 'm+1'
....
n+1 th column the n-Vector from "nm' to "nm+1'
position of 'i' Thus the placement of the i'th number in the sequence can be derived from the above
P(i) = [KJ₀ + _j=1∑ⁿ ((i % m^j)/m^j-1) KJ_j] % m ; i = 1 .. mⁿ
language element <[pos],{v₁},...,{v_n}>
drawback method seem to work only for odd (prime(?)) orders
(reason of which this author currently does not know)

Modular Equations A series of modular equations each depicting a component hypercube
H[_ji] = (_k=0∑^n-1a_k_ki + a_n) % p
due to the effects of the mod operation all values can be limited to radix p numbers
prime digital
p-digital
digit equations p some prime factor of the order m (r = ??)
for powered orders m = q^p (r = pq (?))
p = m (Latin squares) (r = n)
notation r by n+1 matrix (the "latin prescription")
each row multiplies the n+1-Point of each hypercubes position giving the coresponding
component hypercube entry. the amount of needed equations for the described hypercube
depend on the equation type which correspond to the type of component the seperate
equations describe
language element <{a₀},...,{a_r}>
each {a_i} depicts the parameters of each modular equation
note digit changing can be applied on each seperate component independently,
the "latin prescription" does not reflect this general case,
to the language element, in principle, a digit changing permutation can be
attached to each seperate {a_i}

Carpet colorisation
(preliminairy notes) A pattern is repeated throughout the hypercube and a colorisation applied to each copy
The pattern can be used in the first part, so the colorisation of this part can be
the "natural" colorisations thus only need to take place onto the other copies
notation not yet defined

language element <<patrow|...|patrow>,=[perm],..,=[perm]>
note: this LE I think is a square version, some alteration might be needed for
higher dimensioned hypercubes (will decide after implementation)
note The panmagic order 9 investigation of this encyclopedia depicts patterns and augmentative
colorisations with a single number, LE's to handle these I'll decide uppon implementation

formula's functions to depict a hypercube
Some functional descriptions are around, these involves many conditions and are rather
tedious to understand. Besides that they seem only to depict singular hypercubes, none
the less tribute to these efforts for their mathematical importance
latin Squares
(hypercubes (!?)) The panmagic prime order investigation of this encyclopedia show that all
the possible latin squares can be obtained by single parameter formulae,
according to the resulting counting arguments all regular prime order
panmagic square can be with these in combination with digit changing
permutations. This strongly suggest that this construction also is
possible for higher dimensioned hypercubes

models depiction of hypercubes using models
this description needs both a description of the model itself and the distribution of
numbers upon this model. Most common such models is the hypertorus model in n+1 dimension
hyperspace to depict a n dimensional hypercube (so 1 dimension is lost here)

The Hypercube Construction modifiers
Aside from component permutation the various modifiers can be applied onto each described component
this allows us to work with "grand parential component hypercubes (GPGH's)", most discussions need
however need not go that far and remain with the direct results of the basic productions

transposition

^[perm] reordering of the hypercubes axes
The generalisation of the two dimensional "transposition" is a rearangement of the hypercube
axes, which can be easily depicted as a permutation of the n axes of the hypercubes

digit changing

=[perm] changing digits in a component hypercube
The digits in a component hypercube (resulting from a more basic production) might be changed into
an other digit, it is most conveniently depicted by a permutation of the given digits
more hypercubes are reached from the basic production by using this modifier.

main n-agonal permutation

_[perm] permutating the main n-agonal of a component hypercube
each seperate component can of course also be main n-agonally permuted thereby rearanging
every corresponding 1 agonal.

reflection

~refl reflection of a component hypercube in the hypercubes planes
The various components can be reflected in the hypercube planes, therafter the hypercube
must shift back onto its place. The reflection number refl is the sum of 2^axes where
axes are the involved axes in the reflection.

relocation

@[pos] relocating the '0' position in a component hypercube onto position given
Relocating the '0'-position of a component hypercube can of course be done making no assumption
on the components quality

component permutation

#[perm] permuting the components of a hypercube
permutation of the components of a resulting hypercube is of course an action of the entire
set of components and thus an isomorphism on the complete hypercube

Combining hypercubes
Hypercube multiplication and the Hendricks/Trenkler hypercube doubling method are most general methods
of combining two hypercubes into a third, not quite certain yet how to depict the various degrees of freedom
both methods hold (might decide on that upon future date implementation, currently only the formulae are presented)

Hypercube Multiplication Basic Formula ⁿC_m1m2 = ⁿC_m1 * ⁿC_m2 = {ⁿC_m1([ⁿC_m2] - 1) m1ⁿ)}
The basic multiplication formulae might be enough for most purposes.
General Formula ⁿC_m1m2 = (ⁿC_m1;d_m1) * (ⁿC_m2;d_m2) = F { f_ji { (ⁿC_m1;d_m1, ([ⁿC_m2;d_m2]_ji - 1) * m1ⁿ) }}
The general multiplication formula alllows for compensating off sums in the result of a basic multiplication
It allows for multiplication of the non-magic order 2 hypercube to partake in the multiplication theory

Hypercube Doubling ⁿH_2m = ⁿA_2m + G_i,j(ⁿH_m)
with: ⁿA_2m = ⁿT₂(ⁿL_m) = mⁿ (D(F_i,j(ⁿL_m)) - 1) (i,j = (0,1))

The hypercube doubling method can be viewed upon as an application of the general multiplication formula

The Hypercube Construction
Most values for parametrized construction methods will produce hypercubes of low quality
Knight Jump Vectors	Starting from the position of 1 a series of n vectors fill the hypercube
	The position of the '1' in the hypercube is given by a n-Point within the hypercube the step toward the number '2' is given by a n-Vector with entries relative prime to the hypercubes order, after n such steps an occupied place is reached and a second n-Vector is needed too reach a non-occupied place to continue stepping with the first n-Vector thus a total of n n-Vectors are needed to fill up an n-dimensional hypercube.
	notation	n by n+1 matrix (KJ)
	notation	first column the n-Point (position of '1') 2nd column the n-Vector from '1' to '2' 3rd column the n-Vector from 'm' to 'm+1' .... n+1 th column the n-Vector from "nm' to "nm+1'
	position of 'i'	Thus the placement of the i'th number in the sequence can be derived from the above
	position of 'i'	P(i) = [KJ₀ + _j=1∑ⁿ ((i % m^j)/m^j-1) KJ_j] % m ; i = 1 .. mⁿ
	language element	<[pos],{v₁},...,{v_n}>
	drawback	method seem to work only for odd (prime(?)) orders (reason of which this author currently does not know)
Modular Equations	A series of modular equations each depicting a component hypercube
	H[_ji] = (_k=0∑^n-1a_k_ki + a_n) % p
	due to the effects of the mod operation all values can be limited to radix p numbers
	prime digital p-digital digit equations	p some prime factor of the order m (r = ??) for powered orders m = q^p (r = pq (?)) p = m (Latin squares) (r = n)
	notation	r by n+1 matrix (the "latin prescription")
	notation	each row multiplies the n+1-Point of each hypercubes position giving the coresponding component hypercube entry. the amount of needed equations for the described hypercube depend on the equation type which correspond to the type of component the seperate equations describe
	language element	<{a₀},...,{a_r}>
	language element	each {a_i} depicts the parameters of each modular equation
	note	digit changing can be applied on each seperate component independently, the "latin prescription" does not reflect this general case, to the language element, in principle, a digit changing permutation can be attached to each seperate {a_i}
Carpet colorisation (preliminairy notes)	A pattern is repeated throughout the hypercube and a colorisation applied to each copy
	The pattern can be used in the first part, so the colorisation of this part can be the "natural" colorisations thus only need to take place onto the other copies
	notation	not yet defined
	notation
	language element	<<patrow\|...\|patrow>,=[perm],..,=[perm]>
	language element	note: this LE I think is a square version, some alteration might be needed for higher dimensioned hypercubes (will decide after implementation)
	note	The panmagic order 9 investigation of this encyclopedia depicts patterns and augmentative colorisations with a single number, LE's to handle these I'll decide uppon implementation
formula's	functions to depict a hypercube
	Some functional descriptions are around, these involves many conditions and are rather tedious to understand. Besides that they seem only to depict singular hypercubes, none the less tribute to these efforts for their mathematical importance
	latin Squares (hypercubes (!?))	The panmagic prime order investigation of this encyclopedia show that all the possible latin squares can be obtained by single parameter formulae, according to the resulting counting arguments all regular prime order panmagic square can be with these in combination with digit changing permutations. This strongly suggest that this construction also is possible for higher dimensioned hypercubes
models	depiction of hypercubes using models
models	this description needs both a description of the model itself and the distribution of numbers upon this model. Most common such models is the hypertorus model in n+1 dimension hyperspace to depict a n dimensional hypercube (so 1 dimension is lost here)
The Hypercube Construction modifiers
Aside from component permutation the various modifiers can be applied onto each described component this allows us to work with "grand parential component hypercubes (GPGH's)", most discussions need however need not go that far and remain with the direct results of the basic productions
transposition ^[perm]	reordering of the hypercubes axes
transposition ^[perm]	The generalisation of the two dimensional "transposition" is a rearangement of the hypercube axes, which can be easily depicted as a permutation of the n axes of the hypercubes
digit changing =[perm]	changing digits in a component hypercube
digit changing =[perm]	The digits in a component hypercube (resulting from a more basic production) might be changed into an other digit, it is most conveniently depicted by a permutation of the given digits more hypercubes are reached from the basic production by using this modifier.
main n-agonal permutation _[perm]	permutating the main n-agonal of a component hypercube
main n-agonal permutation _[perm]	each seperate component can of course also be main n-agonally permuted thereby rearanging every corresponding 1 agonal.
reflection ~refl	reflection of a component hypercube in the hypercubes planes
reflection ~refl	The various components can be reflected in the hypercube planes, therafter the hypercube must shift back onto its place. The reflection number refl is the sum of 2^axes where axes are the involved axes in the reflection.
relocation @[pos]	relocating the '0' position in a component hypercube onto position given
relocation @[pos]	Relocating the '0'-position of a component hypercube can of course be done making no assumption on the components quality
component permutation #[perm]	permuting the components of a hypercube
component permutation #[perm]	permutation of the components of a resulting hypercube is of course an action of the entire set of components and thus an isomorphism on the complete hypercube
Combining hypercubes
Hypercube multiplication and the Hendricks/Trenkler hypercube doubling method are most general methods of combining two hypercubes into a third, not quite certain yet how to depict the various degrees of freedom both methods hold (might decide on that upon future date implementation, currently only the formulae are presented)
Hypercube Multiplication	Basic Formula	ⁿC_m1m2 = ⁿC_m1 * ⁿC_m2 = {ⁿC_m1([ⁿC_m2] - 1) m1ⁿ)}
	Basic Formula	The basic multiplication formulae might be enough for most purposes.
	General Formula	ⁿC_m1m2 = (ⁿC_m1;d_m1) * (ⁿC_m2;d_m2) = F { f_ji { (ⁿC_m1;d_m1, ([ⁿC_m2;d_m2]_ji - 1) * m1ⁿ) }}
	General Formula	The general multiplication formula alllows for compensating off sums in the result of a basic multiplication It allows for multiplication of the non-magic order 2 hypercube to partake in the multiplication theory
Hypercube Doubling	ⁿH_2m = ⁿA_2m + G_i,j(ⁿH_m) with: ⁿA_2m = ⁿT₂(ⁿL_m) = mⁿ (D(F_i,j(ⁿL_m)) - 1) (i,j = (0,1))
Hypercube Doubling	The hypercube doubling method can be viewed upon as an application of the general multiplication formula