The Magic Encyclopedia ™

The Hypercube
(by Aale de Winkel)

This article deals with the hypercube in general, each n-Point contains a number however no magic condition is imposed yet on the hypercube yet
(this is the subject of the "Magic Hypercube" article)

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'.

ⁿH_m : ⁿ[_ki]_m ε [0..mⁿ-1]

basic
hypercube multiplication ⁿH_m₁ * ⁿH_m₂ :
ⁿ[_ki]_m₁m₂ = ⁿ[ [[_ki \ m₂]_m₁m₁ⁿ]_m₂ + [_ki % m₂]_m₂]_m₁_m₂

Aspects
amount: n! 2ⁿ ⁿH_m^{~R ^[perm(0..n-1)]}
R = _k=0∑^n-1 ((reflect(k)) ? 2^k : 0)
perm(0..n-1) permutation of 0..n-1
The above defines 2ⁿ aspectial variants due to reflection
and n! due to coordinate permutation

Special Hypercubes
The Normal Hypercube ⁿN_m : [_ki] = _k=0∑^n-1 _ki m^k
This hypercube can be seen as the source of all numbers as with
Dynamic Numbering: ⁿH_m = {ⁿN_m} ⁿH_m

The Constant "1" Hypercube ⁿ1_m : [_ki] = 1
This hypercube is usually added to a hypercube to change the
numberrange from analitic [0..mⁿ-1] into regular [1..mⁿ]

Hypercube pecularities
square
cube
tesseract hypercube of dimension 2 (²H_m)
hypercube of dimension 3 (³H_m)
hypercube of dimension 4 (⁴H_m)

Pathfinder
Pf_p a special kind of n-Vector whith only entries -1 0 and 1,
thus used to traverse the hypercubes r-agonals
p = _k=0∑^n-1 (_ki+1) 3^k : Pf_p = <_ki ; 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∑^n-1 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)

r-agonal
polyagonal with r = 1 .. n
the r-dimensional hyperline between oposite corners of the r-dimensional sub-hypercube
the 1-agonal's run parrallel the hypercubes axes
orthogonal
monagonal
diagonal
triagonal
quadragonal 1-agonal
1-agonal
2-agonal
3-agonal
4-agonal

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

1-digital equations
Latin Prescription Matrix
(construction tool) an n by n+1 matrix formed by the coefficients of 1-digital equations
The positions within an hypercube are given by the positions n-Point, 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 left-multiplicated by the given n by n+1 matrix
(if the n-Point containes the regular coordinate symbols [x_i; i=0..n-1] this
regular matrix vector multiplication gives the 1-digital equatons in its original
format, thus the Latin Prescription Matrix is nothing more then 1-digital equations):
N[x_i; i=0..n-1] = [_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

p-digital equations
Latin Prescription Matrix
(construction tool) an np by n+1 matrix formed by the coefficients of p-digital equations
Simular to the 1-digital 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∑^p-1 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 1-digital 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
n-agonal 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 mⁿ 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)

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'.
ⁿH_m : ⁿ[_ki]_m ε [0..mⁿ-1]
basic hypercube multiplication	ⁿH_m₁ * ⁿH_m₂ : ⁿ[_ki]_m₁m₂ = ⁿ[ [[_ki \ m₂]_m₁m₁ⁿ]_m₂ + [_ki % m₂]_m₂]_m₁_m₂
Aspects amount: n! 2ⁿ	ⁿH_m^{~R ^[perm(0..n-1)]} R = _k=0∑^n-1 ((reflect(k)) ? 2^k : 0) perm(0..n-1) permutation of 0..n-1
Aspects amount: n! 2ⁿ	The above defines 2ⁿ aspectial variants due to reflection and n! due to coordinate permutation
Special Hypercubes
The Normal Hypercube	ⁿN_m : [_ki] = _k=0∑^n-1 _ki m^k
The Normal Hypercube	This hypercube can be seen as the source of all numbers as with Dynamic Numbering: ⁿH_m = {ⁿN_m} ⁿH_m
The Constant "1" Hypercube	ⁿ1_m : [_ki] = 1
The Constant "1" Hypercube	This hypercube is usually added to a hypercube to change the numberrange from analitic [0..mⁿ-1] into regular [1..mⁿ]
Hypercube pecularities
square cube tesseract	hypercube of dimension 2 (²H_m) hypercube of dimension 3 (³H_m) hypercube of dimension 4 (⁴H_m)
Pathfinder Pf_p	a special kind of n-Vector whith only entries -1 0 and 1, thus used to traverse the hypercubes r-agonals
Pathfinder Pf_p	p = _k=0∑^n-1 (_ki+1) 3^k : Pf_p = <_ki ; i ε {-1,0,1}>
Latin Hypercube	a Hypercube with only digits (radix order)
Latin Hypercube	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∑^n-1 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)
hyperquadrant	quadrant octant hexadecimant	2 dimensional hyperquadrant (square) 3 dimensional hyperquadrant (cube) 4 dimensional hyperquadrant (tesseract)
r-agonal polyagonal	with r = 1 .. n the r-dimensional hyperline between oposite corners of the r-dimensional sub-hypercube the 1-agonal's run parrallel the hypercubes axes
r-agonal polyagonal	orthogonal monagonal diagonal triagonal quadragonal	1-agonal 1-agonal 2-agonal 3-agonal 4-agonal
The Hypercube construction (general tools)
Knight Jump Matrix (construction tool)	an n-Point juxtaposed to n n-Vectors (describing the Knight Jump construction)
Knight Jump Matrix (construction tool)	The placement of the first number of a sequence of numbers next m numbers are placed an n-Vector from the previous until an occupied place is reached, a further n-Vector is needed to continue the process from a new location the entire hypercube is filled with n such n-Vectors Thus the placement of the i'th number in the sequence can be derived from the above P(i) = [P(0) + _j=0∑^n-1 ((i % m^j+1)/m^j) V_j] % m Whether or not the resulting hypercube is magic depends on the given values
1-digital equations Latin Prescription Matrix (construction tool)	an n by n+1 matrix formed by the coefficients of 1-digital equations
	The positions within an hypercube are given by the positions n-Point, 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 left-multiplicated by the given n by n+1 matrix (if the n-Point containes the regular coordinate symbols [x_i; i=0..n-1] this regular matrix vector multiplication gives the 1-digital equatons in its original format, thus the Latin Prescription Matrix is nothing more then 1-digital equations): N[x_i; i=0..n-1] = [_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
p-digital equations Latin Prescription Matrix (construction tool)	an np by n+1 matrix formed by the coefficients of p-digital equations
	Simular to the 1-digital 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∑^p-1 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 1-digital version
Doubling Method (construction tool)	A general method for doubling any magic hypercube (John R. Hendricks and Marián Trenkler)
Doubling Method (construction tool)	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 n-agonal 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 mⁿ 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)