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 nPoint within the hypercube the step toward the number '2' is given by a nVector with entries relative prime to the hypercubes order, after n such steps an occupied place is reached and a second nVector is needed too reach a nonoccupied place to continue stepping with the first nVector thus a total of n nVectors are needed to fill up an ndimensional hypercube. 

notation  n by n+1 matrix (KJ)  
first column the nPoint (position of '1') 2nd column the nVector from '1' to '2' 3rd column the nVector from 'm' to 'm+1' .... n+1 th column the nVector 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) = [KJ_{0} + _{j=1}∑^{n} ((i % m^{j})/m^{j1}) KJ_{j}] % m ; i = 1 .. m^{n}  
language element  <[pos],{v_{1}},...,{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[_{j}i] = (_{k=0}∑^{n1}a_{k}_{k}i + a_{n}) % p  
due to the effects of the mod operation all values can be limited to radix p numbers  
prime digital pdigital 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+1Point 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  <{a_{0}},...,{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 nagonal permutation _[perm] 
permutating the main nagonal of a component hypercube  
each seperate component can of course also be main nagonally 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  ^{n}C_{m1m2} = ^{n}C_{m1} * ^{n}C_{m2} = {^{n}C_{m1}([^{n}C_{m2}]  1) m1^{n})} 
The basic multiplication formulae might be enough for most purposes.  
General Formula  ^{n}C_{m1m2} = (^{n}C_{m1};d_{m1}) * (^{n}C_{m2};d_{m2}) = F { f_{j}i { (^{n}C_{m1};d_{m1}, ([^{n}C_{m2};d_{m2}]_{j}i  1) * m1^{n}) }}  
The general multiplication formula alllows for compensating off sums in the result of a basic multiplication It allows for multiplication of the nonmagic order 2 hypercube to partake in the multiplication theory 

Hypercube Doubling 
^{n}H_{2m} = ^{n}A_{2m} + G_{i,j}(^{n}H_{m}) with: ^{n}A_{2m} = ^{n}T_{2}(^{n}L_{m}) = m^{n} (D(F_{i,j}(^{n}L_{m}))  1) (i,j = (0,1)) 

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