Magic Hypercube - Aspects

Aspects

A hypercube knows n! 2n Aspectial variants, which are obtained by coordinate reflection ( --> ) and coordinate permutations ( --> i]) effectively giving the Aspectial variant:

nHm~R perm(0..n-1); R = k=0∑n-1 ((reflect(k)) ? 2k : 0) ; perm(0..n-1) a permutation of 0..n-1

Where reflect(k) true iff coordinate k is being reflected, only then 2k is added to R. As is easy to see, only n coordinates can be reflected explaining 2n, the n! permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!

Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in "normal position" by:

= min (by reflection) < ; k = 0..n-2 (by coordinate permutation)

(explicitly stated here: the minimum of all corner points. The axial neighbour sequentially based on axial number)

Read more about this topic:  Magic Hypercube

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