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-1Where 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
Famous quotes containing the word aspects:
“An atheist may be simply one whose faith and love are concentrated on the impersonal aspects of God.”
—Simone Weil (1909–1943)
“I suppose an entire cabinet of shells would be an expression of the whole human mind; a Flora of the whole globe would be so likewise, or a history of beasts; or a painting of all the aspects of the clouds. Everything is significant.”
—Ralph Waldo Emerson (1803–1882)
“All the aspects of this desert are beautiful, whether you behold it in fair weather or foul, or when the sun is just breaking out after a storm, and shining on its moist surface in the distance, it is so white, and pure, and level, and each slight inequality and track is so distinctly revealed; and when your eyes slide off this, they fall on the ocean.”
—Henry David Thoreau (1817–1862)