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: 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. Hendricks called the directions within a hypercubes "pathfinders", these directions are simplest denoted in a ternary number system as:  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:  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:  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. 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).