Hypercube Aspects

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A magic hypercube has many apparently different versions, or aspects. For purposes of enumeration, these are normally considered equivalent solutions, and so are not counted separately.

This page  will discuss the aspects of magic squares, magic cubes, and magic tesseracts. It will also explain the rules for determining which is the normalized version of each.

I have 4 Excel interactive spreadsheets available for download that will visually enhance the understanding of these hypercubes. They are:
Hypercube Generator-3.xls, 2D Aspects.xls, 3D Aspects.xls, and 4D Aspects.xls
Aale de Winkel also discusses hypercube aspects in his Encyclopedia.

Square aspects

In a magic square, there are 4 corners that can be placed in the upper left position. Each of these corners has two rays and because 4 times 2 = 8, a magic square has 8 aspects. 
The example uses order-3 for simplicity. However, because there is only one order-3 magic square, normalization has no practical value.
The number of aspects, however, is the same for any order of magic square. It varies only with the dimension of the hypercube.

Aspects of a magic square

A. = normalized square, B = A. rotated 90°,
C. = A. rotated 180
°, D. = A. rotated 270°,
E. = A. reflected horizontally, F. = A. reflected vertically,
G. = reflected across the right diagonal,
H. = reflected across the leading diagonal.
G. and H. could also be considered rotations of E. and F.

The term basic is often used to indicate that only one of the aspects , the normalized version, is being discussed.

 

Normalized position

For enumerating and listing the magic squares of a given order, one of the eight above positions must be designated the normal position. This standard prevents confusion and permits easy comparison when the list is sorted.
For simplicity, I have used order-3 for the sample of the 8 aspects of a magic square. Of course, normalizing order-3 is meaningless, because there is only one square of this order. The 440 normalized (or basic) magic squares of order-4 were published in 1693 (Frénicle 1605-1675), so such a list has real meaning for this, and higher orders. [1][2]

Interestingly, the number of order-5 magic squares was not established until December, 1973. At that time, Richard Schroeppel calculated and proved that there were 275,305,224 basic order-5 magic squares. [3]
This figure was independently confirmed in 2007 by Walter Trump. [4]

Number of basic magic squares for orders higher then five have only been estimated. [4]

For higher dimension hypercubes, only order-3 numbers are known. There are 4 basic magic cubes, 58 basic magic tesseracts,. 2992 basic magic dimension 5, and 543,328 basic dimension 6 magic hypercubes. [5]

In 1917, W. S. Andrews published all four order-3 cubes, but presumably without realizing that that was all there were. [6]
J. R. Hendricks proved in January 1972 that only four order-3 basic magic cubes were possible. [7].

Hendricks found all 58 basic order-3 magic tesseracts by 1998, and showed that there could be no more. This was confirmed independently by others, including M. Collison, A. W. Johnson, Jr., and Key Ying Lin. [8]

Normalized position – magic squares

Before 1693, Frénicle established two simple rules to determine the standard position for order-4. These same rules are now used for normalizing all other orders of magic squares.

1.      The smallest number in any corner of the magic square must be in the top left corner. If it is not, rotate the square until it is.

2.      The second number in the top row must be of lower then the first number in the second row. These are the two numbers adjacent to the one in the top left corner.

This figure shows the first 4 of the 880 basic magic squares.

The full list of 880 order-4 magic squares is available for downloading from my downloads page.

Text list of the first 4 order-4 magic squares. 

Index   1st row         2nd row           3rd row           4th row
  1   01  02  15  16    12  14  03  05    13  07  10  04    08  11  06  09
  2   01  02  15  16    13  14  03  04    12  07  10  05    08  11  06  09
  3   01  02  16  15    13  14  04  03    12  07  09  06    08  11  05  10
  4   01  03  14  16    10  13  04  07    15  06  11  02    08  12  05  09

Normalized position – magic cubes

The normalized position for magic cubes, tesseracts, etc., is different. Until John Hendricks started working with the higher dimension hypercubes in the second half of the twentieth century, very little consideration had been given to coordinates (or for that matter, lists of solutions).

Because Hendricks work with higher dimensions was so dependent  on coordinates, he decided to go with the conventional geometry. In magic hypercubes, the coordinates are always positive, so the lower, left front corner of the hypercube is the starting point (x, y, z, w,… = 1, 1, 1, 1, …). [8]

The rules for normalizing a magic cube, tesseract, etc. are:

  1. Put the lowest corner number in the lower left front corner.
  2. Then assign the adjacent numbers in increasing magnitude in the x, y, z, … directions

Here are the four basic order-3 magic cubes, normalized, and in increasing index order.

This is the catalogue (list) in sorted order, of the basic order 3 cubes. The cube can be re-constructed with these four numbers and the center number (14). C    X     Y     Z
1   15   17   23
2   15   18   24
4   17   18   26
6   16   17   26

Order-3 magic tesseracts will be discussed on a separate page.
Sufficient to say here that the catalog for the 58 basic tesseracts consists of the five identifying numbers, c, x, y, z, w.

Unfortunately, only order-3 magic hypercubes can be reconstructed from a small group of numbers in a catalogue list. This is because order-3 has only 3 numbers per line, and the center number (which is known), is part of the majority of these lines.

Magic cube aspects

We have seen above that there are 8 different variations (aspects) of the basic magic square.
The 3-dimensional magic cube has many more aspects. In fact, each basic magic cube has 47 apparently different but actually disguised copies.

A cube has 8 corners, each of which may be placed in the lower front left position. Each of these corners has 3 rays which can be labeled in 6 different ways.
This means that there are 8 times 6 = 48 ways that the cube may be viewed!

Magic tesseract aspects

A magic tesseract has 384 aspects because there are 16 corners, each with 4 rays which can be arranged in 24 ways., and 16 x 24 = 384.

No attempt will be made here to illustrate those 384 different aspects. For convenience, I will simply reproduce here an early cube already shown on another page, along with two variations.

Neither of the above are normalized. They are non-normalized aspects of index number 57 (of the 58 order-3 magic tesseracts).

The normalized version has13 in the lower left front corner, with the 4 adjacent numbers 54, 57, 62, 75.

See my order-3 Tesseracts page for a catalogue list of all 58 order-3 magic tesseracts.

To the right is a third aspect (version) of C. Planck's 1905 tesseract. It is shown in Hendricks normalized version.

 

[1] Bernard Frénicle de Bessy , Des Quarrez Magiques. Acad. R. des Sciences 1693 (this is a posthumous paper, not a book)
[2]
Benson, W. & Jacoby, O., New Recreations with Magic Squares, Dover Publ., 1976, 0-486-23236-0
[3] Schroeppel, Richard, Appendix 5: The Order 5 Magic Squares, 1973, 1-16    This is a report of Schroeppels work with enumeration of
     order 5 magic squares , write-up by M. Beeler, mentioned by Gardner in his Scientific American column in January 1976.
[4] Walter Trump http://www.trump.de/magic-squares/,  in an personal email attachment dated April 8, 2007.
[5] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0
[6] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, 419+ pages .
    This is an unaltered reprint of the 1917 Open Court Publication of the second edition.
[7] J. R. Hendricks, The Third Order Magic Cube Complete, Journal of Recreational Mathematics, 5:1:1972:43-50
[8] Hendricks, John R., Magic Squares to Tesseract by Computer, 1998, 0-9684700-0-9
     Hendricks, John R.
, All Third-Order Magic Tesseracts, 1999, 0-9684700-2-5

This page was originally posted November 2007
It was last updated October 20, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz