|
In mathematics, a magic hypercube is the ''k''-dimensional generalization of magic squares, magic cubes and magic tesseracts; that is, a number of integers arranged in an ''n'' × ''n'' × ''n'' × ... × ''n'' pattern such that the sum of the numbers on each pillar (along any axis) as well as the main space diagonals is equal to a single number, the so-called magic constant of the hypercube, denoted ''M''''k''(''n''). It can be shown that if a magic hypercube consists of the numbers 1, 2, ..., ''n''''k'', then it has magic number : For ''n'' = 4, this sequence is . Four-, Five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks. Marian Trenkler proved the following theorem: A ''p''-dimensional magic hypercube of order ''n'' exists if and only if ''p'' > 1 and ''n'' is different from 2 or ''p'' = 1. A construction of a magic hypercube follows from the proof. The R programming language includes a module, library(magic), that will create magic hypercubes of any dimension (with ''n'' a multiple of 4). Change to more modern conventions here-after (basically k ==> n and n ==> m) ==Conventions== It is customary to denote the dimension with the letter 'n' and the order of a hypercube with the letter 'm'. *(''n'') Dimension : the number of directions within a hypercube. *(''m'') Order : the number of numbers along a direction. Further: In this article the analytical number range () is being used. For the regular number range () you can add 1 to each number. This has absolutely no effect on the properties of the hypercube. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Magic hypercube」の詳細全文を読む スポンサード リンク
|