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
nLm
Regularly the used hypercube is a zero n-agonal order m hypercube
(need checking, possibly more general possibilities!?)
Digit changing
Di,j()
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
nA2m
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