The Magic Encyclopedia ™

The Hypercube Doubling Method
{note: investigative article}
(by Aale de Winkel)

in the spring of 1999 <John R Hendricks> came up with a general method for doubling any hypercube. Independently <Marián Trenkler> came up with the same method. This Author merely added a little twist to apply it to the order 2 hypercube as well.

Note: the procedure can be viewed upon as left multiplication of the order m hypercube by the order 2 hypercube. The below formulated can be viewed as samples(?) of the there needed compensations for off sums.

I raised the investigative flag on this article since writing it I noticed some elements which needs some carefull study the below described might have some generalisation. Further the question I raised wheter the augmentation hypercube could be made pan-n-aganal itself.

The Hypercube Doubling Theorem
Given any semi magic latin order m hypercube nLm distributed over the order 2 hypercube nT2
which after changing digits turns into an "augmentation" magic latin hypercube of order 2m
nA2m = nT2(nLm) = mn (D(Fi,j(nLm)) - 1) (i,j = (0,1))
To this augmentation hypercube any order n hypercube can be added to each hyperquadrant such that
nH2m = nA2m + Gi,j(nHm) is a regular magic hypercube
Distribution functions
Fi,j() and Gi,j()
The distribution functions where in the original method mirroring in the central
horizontal and vertical. In order to obtain an order 4 form the order 2 I found
Gi,j() need also to include a center mirroring (NEED TO VERIFY)
latin hypercube
Regularly the used hypercube is a zero n-agonal order m hypercube
(need checking, possibly more general possibilities!?)
Digit changing
Changing digits in the distributed latin hypercube is needed to compensate the
off-sums in the obtained order 2m square. Off sums (+c) in one hyperquadrant
need to be compensated into an adjecent hyperquadrant by a factor (-c) thus
forming the hyperquadrants compenstion patterns (+--+) or (-++-) for squares
and (+--+-++-) or (-++-+--+) for cubes. The encyclopedias database lists the
data for the order 2 and 3 square and cube, and a derivation method to obtain
data for other orders and dimensions as well.
Augmentation hypercube
Multiplying the above obtained square by mn an augmentation hypercube
is obtained to which any order m hypercube can be added, a little care is needed
in the choise of the distribution function Gi,j() to avoid doubly appearing numbers