Magic hypercube
Encyclopedia
In mathematics
, a magic hypercube is the k-dimensional
generalization of magic square
s, magic cube
s and magic tesseract
s; that is, a number of integers arranged in an n × n × n × ... × n pattern such that the sum of the numbers on each pillar (along any axis) as well as the main space diagonal
s is equal to a single number, the so-called magic constant
of the hypercube
, denoted Mk(n). It can be shown that if a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number
If, in addition, the numbers on every cross section
diagonal also sum up to the hypercube's magic number, the hypercube is called a perfect magic hypercube; otherwise, it is called a semiperfect magic hypercube. The number n is called the order of the magic hypercube.
Five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks
.
Marian Trenkler proved the following theorem:
A p-dimensional magic hypercube of order n exists if and only if
p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows from the proof.
The R programming language
includes a module, library(magic), that will create magic hypercubes of any dimension (with n a multiple of 4).
Change to more modern conventions here-after (basically k
with the letter 'n' and the order of a hypercube with the letter 'm'.
Further: In this article the analytical number range [0..mn-1] is being used. For the regular number range [1..mn] you can add 1 to each number. This has absolutely no effect on the properties of the hypercube.
Note: The notation for position can also be used for the value on that position. There where it is appropriate dimension and order can be added to it thus forming: n[ki]m
As is indicated 'k' runs through the dimensions, while the coordinate 'i' runs through all possible values, when values 'i' are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of m, as the magic hypercube resides in n-dimensional modular space.
There can be multiple 'k' between bracket, these can't have the same value, though in undetermined order, which explains the equality of:
[ 1i, kj ] = [ kj, 1i ]
Of course given 'k' also one value 'i' is referred to.
When a specific coordinate value is mentioned the other values can be taken as 0, which is especially the case when the amount of 'k's are limited using pe. #k=1 as in:
[k1 ; #k=1] = [k1 j0 ; #k=1; #j=n-1] ("axial"-neighbor of [k0])
(#j=n-1 can be left unspecified) j now runs through all the values in [0..k-1,k+1..n-1].
Further: without restrictions specified 'k' as well as 'i' run through all possible values, in combinations same letters assume same values. Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section)
Note: as far as I now this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.
Further: "perm(0..n-1)" specifies a permutation
of the n numbers 0..n-1.
[P0, V0 .. Vn-1]
This positions the number 'k' at position:
Pk = P0 + l=0∑n-1((k\ml)%m) Vl; k = 0 .. mn-1.
C. Planck gives in his 1905 article "The theory of Path Nasiks" conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.
This method is also specified by an n by n+1 matrix. However this time it multiplies the n+1 vector [x0,..,xn-1,1], After this multiplication the result is taken modulus m to achieve the n (Latin) hypercubes:
LPk = ( l=0∑n-1 LPk,l xl + LPk,n ) % m
of radix m numbers (also called "digits"). On these LPk's "digit changing" (?i.e. Basic manipulation) are generally applied before these LPk's are combined into the hypercube:
nHm = k=0∑n-1 LPk mk
J.R.Hendricks often uses modular equation, conditions to make hypercubes of various quality can be found on http://www.magichypercubes.com/Encyclopedia at several places (especially p-section)
Both methods fill the hypercube with numbers, the knight-jump guarantees (given appropriate vectors) that every number is present. The Latin prescription only if the components are orthogonal (no two digits occupying the same position)
nHm1 * nHm2 : n[ki]m1m2 = n[ [[ ki \ m2]m1m1n]m2 + [ki % m2]m2]m1m2
Most compounding methods can be viewed as variations of the above, As most qualfiers are invariant under multiplication one can for example place any aspectial variant of nHm2 in the above equation, besides that on the result one can apply a manipulation to improve quality. Thus one can specify pe the J. R. Hendricks / M. Trenklar doubling. These things go beyond the scope of this article.
nHm~R perm(0..n-1); R = k=0∑n-1 ((reflect(k)) ? 2k : 0) ; perm(0..n-1) a permutation of 0..n-1
Where reflect(k) true iff coordinate k is being reflected, only then 2k is added to R.
As is easy to see, only n coordinates can be reflected explaining 2n, the n! permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!
Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in "normal position" by:
[k0] = min([kθ ; θ ε {-1,0}]) (by reflection)
[k1 ; #k=1] < [k+11 ; #k=1] ; k = 0..n-2 (by coordinate permutation)
(explicitly stated here: [k0] the minimum of all corner points. The axial neighbour sequentially based on axial number)
Note:'#' , '^', '_' and '=' are essential part of the notation and used as manipulation selectors.
The term transpose (usually denoted by t) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.
Noted be that reflection is the special case:
~R = _R[n-1,..,0]
Further when all the axes undergo the same ;permutation (R = 2n-1) an n-agonal permutation is achieved, In this special case the 'R' is usually omitted so:
_[perm(0..n-1)] = _(2n-1)[perm(0..n-1)]
Pathfinders
> <ki> ; i ε {-1,0,1}
This gives 3n directions. since every direction is traversed both ways one can limit to the upper half [(3n-1)/2,..,3n-1)] of the full range.
With these pathfinders any line to be summed over (or r-agonal) can be specified:
[ j0 kp lq ; #j=1 #k=r-1 ; k > j ] < j1 kθ l0 ; θ ε {-1,1} > ; p,q ε [0,..,m-1]
which specifies all (broken) r-agonals, p and q ranges could be omitted from this description. The main (unbroken) r-agonals are thus given by the slight modification of the above:
[ j0 k0 l-1 sp ; #j=1 #k+#l=r-1 ; k,l > j ] < j1 k1 l-1 s0 >
nSm = m (mn - 1) / 2.
Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"
Note: This series doesn't start with 0 since a nill-agonal doesn't exist, the numbers correspond with the usual name-calling: 1-agonal = monagonal, 2-agonal = diagonal, 3-agonal = triagonal etc.. Aside from this the number correspond to the amount of "-1" and "1" in the corresponding pathfinder.
In case the hypercube also sum when all the numbers are raised to the power p one gets p-multimagic hypercubes. The above qualifiers are simply prepended onto the p-multimagic qualifier. This defines qualifications as {r-agonal 2-magic}. Here also "2-" is usually replaced by "bi", "3-" by "tri" etc. ("1-magic" would be "monomagic" but "mono" is usually omitted). The sum for p-Multimagic hypercubes can be found by using Faulhaber's formula and divide it by mn-1.
Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the Trump/Boyer {diagonal} cube
is technically seen {1-agonal 2-agonal 3-agonal}.
Nasik magic hypercube gives arguments for using {nasik} as synonymous to {perfect}. The strange generalization of square 'perfect' to using it synonymous to {diagonal} in cubes is however also resolve by putting curly brackets around qualifiers, so {perfect} means {pan r-agonal; r = 1..n} (as mentioned above).
some minor qualifications are:
{ncompact} might be put in notation as : (k)∑ [ji + k1] = 2n nSm / m.
{ncomplete} can simply written as: [ji] + [ji + k(m/2) ; #k=n ] = mn - 1.
Where:
(k)∑ is symbolic for summing all possible k's, there are 2n possibilities for k1.
[ji + k1] expresses [ji] and all its r-agonal neighbors.
for {complete} the complement of [ji] is at position [ji + k(m/2) ; #k=n ].
for squares: {2compact 2complete} is the "modern/alternative qualification" of what Dame Kathleen Ollerenshaw
called most-perfect magic square
, {ncompact ncomplete} is the qualifier for the feature in more than 2 dimensions
Caution: some people seems to equate {compact} with {2compact} instead of {ncompact}. Since this introductory article is not the place to discuss these kind of issues I put in the dimensional pre-superscript n to both these qualifiers (which are defined as shown)
consequences of {ncompact} is that several figures also sum since they can be formed by adding/subtracting order 2 sub-hyper cubes. Issues like these go beyond this articles scope.
This hypercube can be seen as the source of all numbers. A procedure called "Dynamic numbering" makes use of the isomorphism
of every hypercube with this normal, changing the source, changes the hypercube. Usually these sources are limited to direct products of normal hypercubes or normal hyperbeams
(defined as having possibly other orders along the various directions).
The hypercube that is usually added to change the here used "analytic" number range into the "regular" number range. Other constant hypercubes are of course multiples of this one.
, the file format Xml-Hypercubes is developed to describe various hypercubes to ensure human readability as well as programmatical usability. Besides full listings the format offers the ability to invoke mentioned constructions (amongst others)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a magic hypercube is the k-dimensional
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
generalization of magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
s, magic cube
Magic cube
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic...
s and magic tesseract
Magic tesseract
In mathematics, a magic tesseract is the 4-dimensional counterpart of a magic square and magic cube, that is, a number of integers arranged in an n × n × n × n pattern such that the sum of the numbers on each pillar as well as the main space diagonals is equal to a single number,...
s; that is, a number of integers arranged in an n × n × n × ... × n pattern such that the sum of the numbers on each pillar (along any axis) as well as the main space diagonal
Space diagonal
In a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner...
s is equal to a single number, the so-called magic constant
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
of the hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
, denoted Mk(n). It can be shown that if a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number
If, in addition, the numbers on every cross section
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
diagonal also sum up to the hypercube's magic number, the hypercube is called a perfect magic hypercube; otherwise, it is called a semiperfect magic hypercube. The number n is called the order of the magic hypercube.
Five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks
J. R. Hendricks
John Robert Hendricks was a mathematician specializing in magic squares and hypercubes. He has published many articles in the Journal of Recreational Mathematics as well as other Journals.-Early years:...
.
Marian Trenkler proved the following theorem:
A p-dimensional magic hypercube of order n exists if and only if
p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows from the proof.
The R programming language
R (programming language)
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
includes a module, library(magic), that will create magic hypercubes of any dimension (with n a multiple of 4).
Change to more modern conventions here-after (basically k
> n and n
> m)Conventions
It is customary to denote the dimensionDimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
with the letter 'n' and the order of a hypercube with the letter 'm'.
- (n) Dimension : the number of directions within a hypercube.
- (m) Order : the number of numbers along a direction.
Further: In this article the analytical number range [0..mn-1] is being used. For the regular number range [1..mn] you can add 1 to each number. This has absolutely no effect on the properties of the hypercube.
Notations
in order to keep things in hand a special notation was developed:- [ ki; k=[0..n-1]; i=[0..m-1] ]: positions within the hypercube
- < ki; k=[0..n-1]; i=[0..m-1] >: vector through the hypercube
Note: The notation for position can also be used for the value on that position. There where it is appropriate dimension and order can be added to it thus forming: n[ki]m
As is indicated 'k' runs through the dimensions, while the coordinate 'i' runs through all possible values, when values 'i' are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of m, as the magic hypercube resides in n-dimensional modular space.
There can be multiple 'k' between bracket, these can't have the same value, though in undetermined order, which explains the equality of:
[ 1i, kj ] = [ kj, 1i ]
Of course given 'k' also one value 'i' is referred to.
When a specific coordinate value is mentioned the other values can be taken as 0, which is especially the case when the amount of 'k's are limited using pe. #k=1 as in:
[k1 ; #k=1] = [k1 j0 ; #k=1; #j=n-1] ("axial"-neighbor of [k0])
(#j=n-1 can be left unspecified) j now runs through all the values in [0..k-1,k+1..n-1].
Further: without restrictions specified 'k' as well as 'i' run through all possible values, in combinations same letters assume same values. Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section)
Note: as far as I now this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.
Further: "perm(0..n-1)" specifies a permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
of the n numbers 0..n-1.
Construction
Besides more specific constructions two more general construction method are noticeable:KnightJump construction
This construction generalizes the movement of the chessboard horses (vectors <1,2>, <1,-2>, <-1,2>, <-1,-2>) to more general movements (vectors <ki>). The method starts at the position P0 and further numbers are sequentially placed at positions V0 further until (after m steps) a position is reached that is already occupied, a further vector is needed to find the next free position. Thus the method is specified by the n by n+1 matrix:[P0, V0 .. Vn-1]
This positions the number 'k' at position:
Pk = P0 + l=0∑n-1((k\ml)%m) Vl; k = 0 .. mn-1.
C. Planck gives in his 1905 article "The theory of Path Nasiks" conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.
Latin prescription construction
(modular equations).This method is also specified by an n by n+1 matrix. However this time it multiplies the n+1 vector [x0,..,xn-1,1], After this multiplication the result is taken modulus m to achieve the n (Latin) hypercubes:
LPk = ( l=0∑n-1 LPk,l xl + LPk,n ) % m
of radix m numbers (also called "digits"). On these LPk's "digit changing" (?i.e. Basic manipulation) are generally applied before these LPk's are combined into the hypercube:
nHm = k=0∑n-1 LPk mk
J.R.Hendricks often uses modular equation, conditions to make hypercubes of various quality can be found on http://www.magichypercubes.com/Encyclopedia at several places (especially p-section)
Both methods fill the hypercube with numbers, the knight-jump guarantees (given appropriate vectors) that every number is present. The Latin prescription only if the components are orthogonal (no two digits occupying the same position)
Multiplication
Amongst the various ways of compounding, the multiplication can be considered as the most basic of these methods. The basic multiplication is given by:nHm1 * nHm2 : n[ki]m1m2 = n[ [
Most compounding methods can be viewed as variations of the above, As most qualfiers are invariant under multiplication one can for example place any aspectial variant of nHm2 in the above equation, besides that on the result one can apply a manipulation to improve quality. Thus one can specify pe the J. R. Hendricks / M. Trenklar doubling. These things go beyond the scope of this article.
Aspects
A hypercube knows n! 2n Aspectial variants, which are obtained by coordinate reflection ([ki] --> [k(-i)]) and coordinate permutations ([ki] --> [perm[k]i]) effectively giving the Aspectial variant:nHm~R perm(0..n-1); R = k=0∑n-1 ((reflect(k)) ? 2k : 0) ; perm(0..n-1) a permutation of 0..n-1
Where reflect(k) true iff coordinate k is being reflected, only then 2k is added to R.
As is easy to see, only n coordinates can be reflected explaining 2n, the n! permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!
Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in "normal position" by:
[k0] = min([kθ ; θ ε {-1,0}]) (by reflection)
[k1 ; #k=1] < [k+11 ; #k=1] ; k = 0..n-2 (by coordinate permutation)
(explicitly stated here: [k0] the minimum of all corner points. The axial neighbour sequentially based on axial number)
Basic manipulations
Besides more specific manipulations, the following are of more general nature- #[perm(0..n-1)] : component permutation
- ^[perm(0..n-1)] : coordinate permutation (n 2: transpose)
- _2axis[perm(0..m-1)] : monagonal permutation (axis ε [0..n-1])
- =[perm(0..m-1)] : digit change
Note:
Component permutation
Defined as the exchange of components, thus varying the factor mk in mperm(k), because there are n component hypercubes the permutation is over these n componentsCoordinate permutation
The exchange of coordinate [ki] into [perm(k)i], because of n coordinates a permutation over these n directions is required.The term transpose (usually denoted by t) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.
Monagonal permutation
Defined as the change of [ki] into [kperm(i)] alongside the given "axial"-direction. Equal permutation along various axes can be combined by adding the factors 2axis. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding permutation of m numbers.Noted be that reflection is the special case:
~R = _R[n-1,..,0]
Further when all the axes undergo the same ;permutation (R = 2n-1) an n-agonal permutation is achieved, In this special case the 'R' is usually omitted so:
_[perm(0..n-1)] = _(2n-1)[perm(0..n-1)]
Digitchanging
Usually being applied at component level and can be seen as given by [ki] in perm([ki]) since a component is filled with radix m digits, a permutation over m numbers is an appropriate manner to denote these.Pathfinders
J. R. Hendricks called the directions within a hypercubes "pathfinders", these directions are simplest denoted in a ternary number system as:
Pfp where: p = k=0∑n-1 (ki + 1) 3k <
> <ki> ; i ε {-1,0,1}This gives 3n directions. since every direction is traversed both ways one can limit to the upper half [(3n-1)/2,..,3n-1)] of the full range.
With these pathfinders any line to be summed over (or r-agonal) can be specified:
[ j0 kp lq ; #j=1 #k=r-1 ; k > j ] < j1 kθ l0 ; θ ε {-1,1} > ; p,q ε [0,..,m-1]
which specifies all (broken) r-agonals, p and q ranges could be omitted from this description. The main (unbroken) r-agonals are thus given by the slight modification of the above:
[ j0 k0 l-1 sp ; #j=1 #k+#l=r-1 ; k,l > j ] < j1 k1 l-1 s0 >
Qualifications
A hypercube nHm with numbers in the analytical numberrange [0..mn-1] has the magic sum:nSm = m (mn - 1) / 2.
Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"
- {r-agonal} : all main (unbroken) r-agonals are summing.
- {pan r-agonal} : all (unbroken and broken) r-agonals are summing.
- {magic} : {1-agonal n-agonal}
- {perfect} : {pan r-agonal; r = 1..n}
Note: This series doesn't start with 0 since a nill-agonal doesn't exist, the numbers correspond with the usual name-calling: 1-agonal = monagonal, 2-agonal = diagonal, 3-agonal = triagonal etc.. Aside from this the number correspond to the amount of "-1" and "1" in the corresponding pathfinder.
In case the hypercube also sum when all the numbers are raised to the power p one gets p-multimagic hypercubes. The above qualifiers are simply prepended onto the p-multimagic qualifier. This defines qualifications as {r-agonal 2-magic}. Here also "2-" is usually replaced by "bi", "3-" by "tri" etc. ("1-magic" would be "monomagic" but "mono" is usually omitted). The sum for p-Multimagic hypercubes can be found by using Faulhaber's formula and divide it by mn-1.
Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the Trump/Boyer {diagonal} cube
Perfect magic cube
In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant....
is technically seen {1-agonal 2-agonal 3-agonal}.
Nasik magic hypercube gives arguments for using {nasik} as synonymous to {perfect}. The strange generalization of square 'perfect' to using it synonymous to {diagonal} in cubes is however also resolve by putting curly brackets around qualifiers, so {perfect} means {pan r-agonal; r = 1..n} (as mentioned above).
some minor qualifications are:
- {ncompact} : {all order 2 subhyper cubes sum to 2n nSm / m}
- {ncomplete} : {all pairs halve an n-agonal apart sum equal (to (mn - 1)}
{ncompact} might be put in notation as : (k)∑ [ji + k1] = 2n nSm / m.
{ncomplete} can simply written as: [ji] + [ji + k(m/2) ; #k=n ] = mn - 1.
Where:
(k)∑ is symbolic for summing all possible k's, there are 2n possibilities for k1.
[ji + k1] expresses [ji] and all its r-agonal neighbors.
for {complete} the complement of [ji] is at position [ji + k(m/2) ; #k=n ].
for squares: {2compact 2complete} is the "modern/alternative qualification" of what Dame Kathleen Ollerenshaw
Kathleen Ollerenshaw
Dame Kathleen Mary Ollerenshaw, née Timpson, DBE is a British mathematician and politician. Deaf since the age of eight, she loved doing arithmetic problems as a child. As a young woman, she attended St Leonards School and Sixth Form College in St Andrews, Scotland where today the house of young...
called most-perfect magic square
Most-perfect magic square
A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties:# Each 2×2 subsquare sums to 2s, where s = n2 + 1....
, {ncompact ncomplete} is the qualifier for the feature in more than 2 dimensions
Caution: some people seems to equate {compact} with {2compact} instead of {ncompact}. Since this introductory article is not the place to discuss these kind of issues I put in the dimensional pre-superscript n to both these qualifiers (which are defined as shown)
consequences of {ncompact} is that several figures also sum since they can be formed by adding/subtracting order 2 sub-hyper cubes. Issues like these go beyond this articles scope.
The "normal hypercube"
nNm : [ki] = k=0∑n-1 ki mkThis hypercube can be seen as the source of all numbers. A procedure called "Dynamic numbering" makes use of the isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
of every hypercube with this normal, changing the source, changes the hypercube. Usually these sources are limited to direct products of normal hypercubes or normal hyperbeams
Magic hyperbeam
A magic hyperbeam is a variation on a magic hypercube where the orders along each direction may be different. As such a magic hyperbeam generalises the two dimensional magic rectangle and the three dimensional magic beam, a series that mimics the series magic square, magic cube and magic hypercube...
(defined as having possibly other orders along the various directions).
The "constant 1"
n1m : [ki] = 1The hypercube that is usually added to change the here used "analytic" number range into the "regular" number range. Other constant hypercubes are of course multiples of this one.
See also
- Magic hyperbeamMagic hyperbeamA magic hyperbeam is a variation on a magic hypercube where the orders along each direction may be different. As such a magic hyperbeam generalises the two dimensional magic rectangle and the three dimensional magic beam, a series that mimics the series magic square, magic cube and magic hypercube...
- Nasik magic hypercube
- Space diagonalSpace diagonalIn a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner...
- John R. Hendricks
File format
Based on XMLXML
Extensible Markup Language is a set of rules for encoding documents in machine-readable form. It is defined in the XML 1.0 Specification produced by the W3C, and several other related specifications, all gratis open standards....
, the file format Xml-Hypercubes is developed to describe various hypercubes to ensure human readability as well as programmatical usability. Besides full listings the format offers the ability to invoke mentioned constructions (amongst others)
External links
- The Magic Encyclopedia Articles by Aale de Winkel
- Magic Cubes - Introduction by Harvey D. Heinz
- Magic Cubes and Hypercubes - References Collected by Marian Trenkler
- An algorithm for making magic cubes by Marian Trenkler
- multimagie.com Articles by Christian Boyer
Further reading
- J.R.Hendricks: Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9
- Planck, C., M.A.,M.R.C.S., The Theory of Paths Nasik, 1905, printed for private circulation. Introductory letter to the paper