The common sum is called the magic constant of the hypercube, and is sometimes denoted Mk(n).
Four-, 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 nasik magic hypercube has 1/2(3n − 1) lines of m numbers passing through each of the mn cells.
In the introductory to his paper, he wrote; Analogy suggest that in the higher dimensions we ought to employ the term nasik as implying the existence of magic summations parallel to any diagonal, and not restrict it to diagonals in sections parallel to the plane faces.
The term is used in this wider sense throughout the present paper.In 1917, Dr. Planck wrote again on this subject.
It is not difficult to perceive that if we push the Nasik analogy to higher dimensions the number of magic directions through any cell of a k-fold must be ½(3k-1).In 1939, B. Rosser and R. J. Walker published a series of papers on diabolic (perfect) magic squares and cubes.
They specifically mentioned that these cubes contained 13m2 correctly summing lines.
Then, 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 brackets, these cannot have the same value, though in undetermined order, which explains the equality of:
Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section) Note: as far as I know this notation is not in general use yet(?
Thus the method is specified by the n by n+1 matrix: This positions the number 'k' at position: C. Planck gives in his 1905 article "The theory of Path Nasiks" conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.
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: of radix m numbers (also called "digits").
The Latin prescription only if the components are orthogonal (no two digits occupying the same position) Amongst the various ways of compounding, the multiplication[8] can be considered as the most basic of these methods.
The basic multiplication is given by: Most compounding methods can be viewed as variations of the above, As most qualifiers are invariant under multiplication one can for example place any aspectual variant of nHm2 in the above equation, besides that on the result one can apply a manipulation to improve quality.
permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!
Thus any hypercube can be represented shown in "normal position" by: (explicitly stated here: [k0] the minimum of all corner points.
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: 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: 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.
since every direction is traversed both ways one can limit to the upper half [(3n-1)/2,..,3n-1)] of the full range.
The main (unbroken) r-agonals are thus given by the slight modification of the above: A hypercube nHm with numbers in the analytical numberrange [0..mn-1] has the magic sum: 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.
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}.
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).
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.
There where it is appropriate dimension and orders can be added to it thus forming: n[ki]m0,..,mn-1 Description of more general methods might be put here, I don't often create hyperbeams, so I don't know whether Knightjump or Latin Prescription work here.
This is with the exception of mk=1 of course, which allows for general identities like: Which goes beyond the scope of this introductory article since any number has but one complement only one of the directions can have mk = 2.
The exchange of coördinaat [ki] into [perm(k)i], because of n coördinates 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 "coördinaatpermutation" might be preferable.
Basic multiplications of normal hyperbeams play a special role with the "Dynamic numbering" of magic hypercubes of order k=0Πn-1 mk.