The Magic Hyperbeam  

The hyperbeam is qualified as magic if all monagonals sum to the same number relative to the length of the axes, 'these magic sums are:' S_{i} = m_{i} (_{j=0}∏^{n1}m_{j}  1) / 2 The bodys "wraparound" nagonal to sum to its magic sum: S = lcm(m_{i} ; i = 1 .. n) (_{j=0}∏^{n1}m_{j}  1) / 2 if the numbers m_{i} are all relatively prime this sum reaches its maximum S_{max} = _{j=0}∏^{n1}m_{j} (_{j=0}∏^{n1}m_{j}  1) / 2 

Even / Odd Theorem 
there are no mixed even / odd orders in a magic hyperbeam except of course m_{k} = 1 

when one of the m_{k}'s is even the magic sum's product is even so the only way the magic sums are integers is when all m_{k}'s are even 

{magic}  only the monagonals sum up to their magic sum  
{strictmagic}  the bodys wraparound nagonals do sum up to their magic sum  
{pseudomagic} 
monagonals either sum to the magic constant or form a consecutive array of sums in arithmetic progression based on a single constant 

a second version of pseudo magicness is when a single direction show these sums starting at a different position 

Qualifiers are less developped for the hyperbeam as for the hypercube complying with most authors {magic} for the hyperbeam is defined for the monagonals only while {strictmagic} is the one that correspond to the hypercube meaning of {magic} when all orders m_{k} turn equal, while {magic} corresponds with {semimagic} 

The Magic Hyperbeam creation  
As to my Knowledge Magic Hyperbeams are created by ad hoc methods The following describes a few "compensating functions" to hyperbeam products which where used in the MagicHyperbeam Examples Page 

Monogonal Shift (Even/Odd): _{k}Shift_{[li]} 
_{k}Shift_{[li]} : [_{k}j] <= [(_{l}i * m_{k}) + _{k}j]  
This Monagonal Shift shifts the kaxes values of the [_{l}i] "plane"  
Preparation (Even) Prep(D)_{[ni]} 
Prep(D)_{[ni]}: ((D[_{n}i]==1) ? P _{n}i + ^{n1}B_{(m..)} : P (_{n}i + 1)  1  ^{n1}B_{(m..)}) ; P = _{k=0}∏^{n1}m_{k} D[_{n}i] ε {1,1} D[_{n}i] = m_{n}  1  D[_{n}i] ; _{i=0}∑^{mn1} D[_{n}i] = 0 

This prepares a doubly even dimensional growth for swapping  
Swap (Even)  Swap: [_{n}i] <= [(((_{k=0}∑^{n1} _{k}i) % 2 == 0) ? 1 : 1) * _{n}i]  
This reflects half the monagonals dimensional grown and prepped 