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His interest in magic squares led to higher dimensions: magic cubes, tesseracts, etc. He developed a new diagram for the four-dimensional tesseract. This was published in 1962 when he showed constructions of four-, five-, and six-dimensional magic hypercubes of order three. [1]
Proper: A proper magic cube is a magic cube belonging to one of the six classes of magic cube, but containing exactly the minimum requirements for that class of cube. i.e. a proper simple or pantriagonal magic cube would contain no magic squares, a proper diagonal magic cube would contain exactly 3m + 6 simple magic squares, etc. This term was ...
Pandiagonal magic cubes are extensions of diagonal magic cubes (in which only the unbroken diagonals need to have the same sum as the rows of the cube) and generalize pandiagonal magic squares to three dimensions. In a pandiagonal magic cube, all 3m planar arrays must be panmagic squares. The 6 oblique squares are always magic. Several of them ...
A pandiagonal magic square remains pandiagonally magic not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an n × n {\displaystyle n\times n} pandiagonal magic square can be regarded as having 8 n 2 {\displaystyle 8n^{2}} orientations.
In a magic cube, a broken space diagonal is a sequence of cells of the cube that follows a line parallel to a space diagonal of the cube, and continues on the corresponding point of an opposite face whenever it reaches a face of the cube. [1] [2] The corresponding concept in two-dimensional magic squares is a broken diagonal.
A magic hyperbeam (n-dimensional magic rectangle) is a variation on a magic hypercube where the orders along each direction may be different. As such a magic hyperbeam generalises the two dimensional magic rectangle and the three dimensional magic beam, a series that mimics the series magic square, magic cube and magic hypercube.