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:
“The happiest two-job marriages I saw during my research were ones in which men and women shared the housework and parenting. What couples called good communication often meant that they were good at saying thanks to one another for small aspects of taking care of the family. Making it to the school play, helping a child read, cooking dinner in good spirit, remembering the grocery list,... these were silver and gold of the marital exchange.”
—Arlie Hochschild (20th century)
“The power of a text is different when it is read from when it is copied out.... Only the copied text thus commands the soul of him who is occupied with it, whereas the mere reader never discovers the new aspects of his inner self that are opened by the text, that road cut through the interior jungle forever closing behind it: because the reader follows the movement of his mind in the free flight of day-dreaming, whereas the copier submits it to command.”
—Walter Benjamin (1892–1940)
“Grammar is a tricky, inconsistent thing. Being the backbone of speech and writing, it should, we think, be eminently logical, make perfect sense, like the human skeleton. But, of course, the skeleton is arbitrary, too. Why twelve pairs of ribs rather than eleven or thirteen? Why thirty-two teeth? It has something to do with evolution and functionalism—but only sometimes, not always. So there are aspects of grammar that make good, logical sense, and others that do not.”
—John Simon (b. 1925)