Transformations  

The following transformations play an important part with especially pandiagonal odd order magic square, the listed transforms can be used to transform the set of "essentially different" square onto the set of "different" panmagic squares. For order 5 this cuts down the total amount of panmagic squares (144) by a factor 4 to 36 essentially different magic squares other orders have corresponding factors (yet to be determined), the formulae do not only work for prime orders but also for noneprime orders, even when parameters are not relatively prime, in which case continuation might might be quite arbitrarily defined (properties of which are still under investigation, since the properties of the samples squares seems to blur things. (The intent is to make this discussion feature independent)) 

all order transforms  
the panflip 
The reflection of the hypercube into the 0plane followed by a pan relocation of the origins 0plane onto the targets 0plane is revered to as a panflip in the panmagic square articles (0plane is in the above generalisation relative to the chosen 1agonal (relavant coordinate is '0')) 

Odd order tranforms  
diagonalrow transform R+ or R 
The transformation of the main dagonal into a row  
With these transforms the diagonal is mapped onto the top row by: R+: [i + j,i  j] > [i,j] i,j = 0..m1 (modular m) R: [i  j,i + j] > [i,j] i,j = 0..m1 (modular m) where the formula also show all the original square subdiagonals to be transformed into colums in either direction, this can be easily seen to be only possible with odd orders 

diagonalcolumn transform C+ or C 
The transformation of the main dagonal into a column  
With these transforms the diagonal is mapped onto the left column by: C+: [i + j,i  j] > [j,i] i,j = 0..m1 (modular m) C: [i  j,i + j] > [j,i] i,j = 0..m1 (modular m) where the formula also show all the original square subdiagonals to be transformed into colums in either direction, this can be easily seen to be only possible with odd orders 

C+ and C are related to one another by the horizontal panflip, moving the first column of the above C+ to the centercolumn gives the coördinate transform C+_{q} = [(q  x + y) % m,(q + 1 + x + y) % m] (with 2 q + 1 = m) applying this twice gives the coördinate transform C+_{q}^{2} = [(q + 1 + 2x) % m,(q + 1 + 2y) % m] more commenly refered to as the symmetrical diagonal permutation [ (q + 1 + 2 x) % m ; q = (m1)/2 ; x = 0 .. m1 ] R+ and R are related to C+ and C by transposition so same thing applies 

subdiagonal changing permutation P_{a} 
Special main diagonal permutation to change the subdiagonal  
Special diagonal permutations can be seen to change subdiagonals in square of odd order, these transforms can be seen to be given by [a * i % m; i = 0 .. m1; a = 1 .. m1] 

nagonal1agonal transform  a generalisation of the diagonal row transform  
As a generalisation of the diagonal row transform the hypercubes nagonal is mapped onto a hypercube 1agonal, from each number each broken sub nagonal is traced and mappen sequaentially on a still free 1agonal, thus one obtains a new hypercube which preserve the panmagic quality (I curently believe that it doesn't really matter which subdiagonal is place upon which 1agonal one merely obtains a transpositional equivalent which when normalized gives the same hypercube) NOTE: this a\ony works with odd orders because the sub nagonals only intersect the main nagonal once. 

Doubly even orders have some special transformations to make magic hypercubes from the most basic hypercube (or other generating hypercubes) 

Pandiagonal transform P(..) 
The pandiagonal transform is defined as a 2 step process on hypercubes of order 4p 

[_{j}(2p+k)] <> [_{j}(4p1k)] (j = 0..n1; k = 0..p1 [_{j}k] <> [_{j}(2p+k)] (j = 0..n1; (_{j}∑^{n1}[_{j}k] ) % 2 = 1; k < 2p) 

The first step of the process is a mirrorswap of the third and fourth quater of each 1agonal. The second step is a swap of alternate 1agonal numbers with a number 2m apart on the same 1agonal. In overall the first step compensate the 1agonal off sums of the numbers modulo (2m)^{2}, while the second swap compensate the part of the off sums due to the high component of the numbers note: the transform does not compensate diagonal sums, so the diagonals need sum already to the magic sum, for the result to be diagonal thus the origin need to be "pan antisemimagic" or "pan blockmagic" (must take a closer look at these qualifications) These conditions are met by the normal square, and left multiplications with N_{2} or N_{2}^{t} of either order 2p normal square or order 2p panmagic square. 

In reverse given a position [_{j}k] coordinates when summing to an odd number all coordinates swap with coordinates 2p further if (_{j}∑^{n1}[_{j}k] ) % 2 = 1 then [_{j}k] <> [(_{j}k + 2p) % 4p] position coordinates bigger (or equal) then 2p swap with its 3p mirror if [_{j}k] >= 2p then [_{j}k] <> [6p  _{j}k] The thus obtained position is the position within the Generating Hypercube corresponding with the generated hypercubes position. 

biPandiagonal transform biP(..) 
The bipandiagonal transform is defined as a 2 step process on hypercubes of order 4p 

[_{j}k] <> [_{(n1j)}(4p1k)]
(j = 0..n2 ; k = "odd" ; k < 2p) (j = 0..n2 ; k = "even" ; k >= 2p) Note: the above might be slightly off, as also I but recently noted the the previous "odd" condition with P(..) altered in the above. 

The biPandiagonal transform swaps every other number with the hypercubes center mirror positioned the result of this transformation seems to be a magic square every other panrelocation hence the name 