The Hypercube  

The hypercube consists of squares of numbers, the amount of numbers along any monagonal is the same in all directions. This is called the order and denoted by 'm', while the dimension is denoted by 'n'. 

^{n}H_{m} : ^{n}[_{k}i]_{m} ε [0..m^{n}1]  
basic hypercube multiplication 
^{n}H_{m1} * ^{n}H_{m2} : ^{n}[_{k}i]_{m1m2} = ^{n}[ [[_{k}i \ m_{2}]_{m1}m_{1}^{n}]_{m2} + [_{k}i % m_{2}]_{m2}]_{m1}_{m2} 

Aspects amount: n! 2^{n} 
^{n}H_{m}^{~R ^[perm(0..n1)]} R = _{k=0}∑^{n1} ((reflect(k)) ? 2^{k} : 0) perm(0..n1) permutation of 0..n1 

The above defines 2^{n} aspectial variants due to reflection and n! due to coordinate permutation 

Special Hypercubes  
The Normal Hypercube  ^{n}N_{m} : [_{k}i] = _{k=0}∑^{n1} _{k}i m^{k}  
This hypercube can be seen as the source of all numbers as with Dynamic Numbering: ^{n}H_{m} = {^{n}N_{m}} ^{n}H_{m} 

The Constant "1" Hypercube  ^{n}1_{m} : [_{k}i] = 1  
This hypercube is usually added to a hypercube to change the numberrange from analitic [0..m^{n}1] into regular [1..m^{n}] 

Hypercube pecularities  
square cube tesseract 
hypercube of dimension 2 (^{2}H_{m}) hypercube of dimension 3 (^{3}H_{m}) hypercube of dimension 4 (^{4}H_{m}) 

Pathfinder Pf_{p} 
a special kind of nVector whith only entries 1 0 and 1, thus used to traverse the hypercubes ragonals 

p = _{k=0}∑^{n1} (_{k}i+1) 3^{k} : Pf_{p} = <_{k}i ; i ε {1,0,1}>  
Latin Hypercube  a Hypercube with only digits (radix order)  
The latin hypercube is generally seen as a component of an hypercube The latin hypercube comes about if all the numbers are put into the given by the order, ie each number N = _{j=0}∑^{n1} n_{j} m^{j} the hypercube formed by the numbers n_{j} form a latin component hypercube. In reverse given a set of latin hypercubes a regular hypercube can be formed, the quality of that hypercube of course depend on the latin hypercubes used. (most commonly one gets irregular or generalized hypercubes) 

hyperquadrant 
the n dimensional subfigure with half the hypercube order in all directions usually situated at one of the hypercubes corners, hyperquadrants might overlap (odd order) 

quadrant octant hexadecimant 
2 dimensional hyperquadrant (square) 3 dimensional hyperquadrant (cube) 4 dimensional hyperquadrant (tesseract) 

ragonal polyagonal 
with r = 1 .. n the rdimensional hyperline between oposite corners of the rdimensional subhypercube the 1agonal's run parrallel the hypercubes axes 

orthogonal monagonal diagonal triagonal quadragonal 
1agonal 1agonal 2agonal 3agonal 4agonal 

The Hypercube construction (general tools)  
Knight Jump Matrix (construction tool) 
an nPoint juxtaposed to n nVectors (describing the Knight Jump construction)  
The placement of the first number of a sequence of numbers next m numbers are placed an nVector from the previous until an occupied place is reached, a further nVector is needed to continue the process from a new location the entire hypercube is filled with n such nVectors Thus the placement of the i'th number in the sequence can be derived from the above P(i) = [P(0) + _{j=0}∑^{n1} ((i % m^{j+1})/m^{j}) V_{j}] % m Whether or not the resulting hypercube is magic depends on the given values 

1digital equations Latin Prescription Matrix (construction tool) 
an n by n+1 matrix formed by the coefficients of 1digital equations  
The positions within an hypercube are given by the positions nPoint, adding a 1 towards its end makes it into a (n+1)Point, this however is merely a technical necessity to allow it to be leftmultiplicated by the given n by n+1 matrix (if the nPoint containes the regular coordinate symbols [x_{i}; i=0..n1] this regular matrix vector multiplication gives the 1digital equatons in its original format, thus the Latin Prescription Matrix is nothing more then 1digital equations): N[x_{i}; i=0..n1] = [_{j=0}∑^{n} LP_{i}^{j} x_{j}] % m Note: that these equations are taken modulo m since LP's are used to obtain Latin Hypercubes which later combine into hypercubes Whether or not the resulting hypercube is magic depends on the given values 

pdigital equations Latin Prescription Matrix (construction tool) 
an np by n+1 matrix formed by the coefficients of pdigital equations  
Simular to the 1digital equation version but works only for orders of type m^{p} In order to obtain the extra equations the coordinates are split into p parts base m such that: x_{i} = _{j=0}∑^{p1} y_{j} m^{j} this p coordinate parts, can be multiplicated (in a special manner (not quite regular)) into np modular equations like the one given in the 1digital version 

Doubling Method (construction tool) 
A general method for doubling any magic hypercube (John R. Hendricks and Marián Trenkler) 

Independently from each other (as reported to this author) John R. Hendricks and Marián Trenkler discovered a general method for doubling any hypercube. A zero nagonal order m hypercube is distributed onto each hyperquadrant, digit changes are applied to make all sums of this "doubling component" sum to the same number thereafter each number is multilied by m^{n} any hypercube order m can now be added to each hyperquadrant. (This is a mere qualitative description of the method, some degrees of freedom are seen by this author but not yet quantisized) 