The Magic Encyclopedia ™

The Magic Hyperbeam
(by Aale de Winkel)

The Magic Hyperbeam is defined as a hyperbeam, filled with integers [0..j=0n-1mj-1] (analitic number range) and can be seen as a magic hypercube where all the squares are replaced by rectangles, because of the different amount of numbers on each 1-agonal each 1-agonal sums to a different sum. Complying with the qualification of calling rectangles magic when only the monagonals sum those hyperbeams with correctly summing n-agonals I qualify "strict-magic" this of course only makes sense when not all orders are relatively prime, in which case all numbers are summed over.

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:'
Si = mi (j=0n-1mj - 1) / 2
The bodys "wrap-around" n-agonal to sum to its magic sum:
S = lcm(mi ; i = 1 .. n) (j=0n-1mj - 1) / 2
if the numbers mi are all relatively prime this sum reaches its maximum
Smax = j=0n-1mj (j=0n-1mj - 1) / 2
Even / Odd Theorem there are no mixed even / odd orders in a magic hyperbeam
except of course mk = 1
when one of the mk's is even the magic sum's product is even
so the only way the magic sums are integers is when all mk's are even
{magic} only the monagonals sum up to their magic sum
{strict-magic} the bodys wrap-around n-agonals do sum up to their magic sum
{pseudo-magic} 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 {strict-magic} is the one that correspond to the hypercube meaning of {magic} when
all orders mk turn equal, while {magic} corresponds with {semi-magic}
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):
kShift[li]
kShift[li] : [kj] <= [(li * mk) + kj]
This Monagonal Shift shifts the k-axes values of the [li] "plane"
Preparation (Even)
Prep(D)[ni]
Prep(D)[ni]: ((D[ni]==1) ? P ni + n-1B(m..) : P (ni + 1) - 1 - n-1B(m..)) ; P = k=0n-1mk
D[ni] ε {-1,1} D[ni] = mn - 1 - D[ni] ; i=0mn-1 D[ni] = 0
This prepares a doubly even dimensional growth for swapping
Swap (Even) Swap: [ni] <= [(((k=0n-1 ki) % 2 == 0) ? 1 : -1) * ni]
This reflects half the monagonals dimensional grown and prepped