Hypercube Classes

Introduction       Magic Squares      Magic Cubes      Magic Tesseracts

Introduction
Magic hypercubes may be classified by reference to which r-agonals sum correctly to the magic constant. For magic squares and magic cubes, this has been discussed previously. [1][2][3][4]
A brief review will be presented on this page, using a different prospective. Then I will present the classes for the magic tesseract. Note that within these classes, the hypercube may have additional features, such as associated, compact complete, inlaid, multiply, etc.

On this page, the variable r will range from 1 to m, where m indicates the order. n, as usual on my pages, will indicate the dimension of the hypercube.
Another name for  1-agonal is orthogonal lines (those parallel to the edges of the hypercube). Only 1 coordinate change when moving along the line.
Another name for 2-agonal is diagonal. 2 coordinates change when moving along the line.
Another name for 3-agonal is triagonal. 3 coordinates change when moving along the line.
Another name for 4-agonal is quadragonal. 4 coordinates change when moving along the line.
A particular hypercube may have some, but not all of the correct r-agonals that would qualify it for a higher class.

The prefix pan indicates all of that r-agonal, both 1-segment and multi-segment (broken).

A pandiagonal magic square may be transformed to another pandiagonal magic square by moving a row or column from one side of the square to the opposite side. Similarly, a pantriagonal magic cube may be transformed into another pantriagonal magic cube by moving a plane from one side of the cube to the other! Furthermore, a panquadragonal magic tesseract may be transformed to another one by moving a cube from one side to the other! Etc.

Magic squares (n = 2)

There are only two classes of magic squares.

Magic cubes (n = 3)

There are six classes of magic cubes.

Magic tesseracts (n = 4)

There are 2 classes of magic squares and 6 classes of magic cubes. So it is to be expected that there will be many more classes of magic tesseract. In fact, there are 18. Names of these are arbitrarily chosen to be descriptive, rather then concise. As in the case for the square and the cube, these classes are listed in order of increasing number of correct lines. In late 2007, Mitsutoshi Nakamura had constructed a tesseract in most of these classes. [4]

 Statistical information on these classifications are shown in tables on my Hypercube Math page.
I have not personally checked most of the tesseracts mentioned in this table.

Footnotes
[1] Heinz & Hendricks, A Unified Classification System for Magic Hypercubes, Journal of Recreational Mathematics, 32:1, 2003-2004, pages 30-36
[2] H. D. Heinz, The First (?) Perfect Magic Cubes, JRM, 33:2, 2004-2005, pages 116-119
[3] My Six Classes of Cubes http://members.shaw.ca/hdhcubes/#6 Classes of Cubes
[4] Mitsutoshi Nakamura’s http://homepage2.nifty.com/googol/magcube/en/classes.htm   
[5] Nasik               from Frost’s original term for pandiagonal magic squares, and Planck’s subsequent expansion of the definition to include the highest class of all dimensions of magic hypercubes. See [6][7][8] and my Nasik article
[6] A. H. Frost, Invention of Magic Cubes, Quarterly Journal of Mathematics, 7, 1866, pages 92-102. See page 99, para. 23 and page 100, para. 26.
[7] C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England),1905
 (Available from The University Library, Cambridge).
[8] W. S. Andrews, Magic Squares and Cubes. Open Court Publ.,1917. Pages 365,366, by Dr. C. Planck.  Re-published by Dover Publ., 1960, Pages 365,366 (no ISBN); Dover Publ., 2000, 0486206580; Cosimo Classics, Inc., 2004, 1596050373

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Harvey Heinz   harveyheinz@shaw.ca
This page originated November, 2007
This page last updated October 14, 2009
Copyright © 2004 by Harvey D. Heinz