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 Generator3.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 order3 for simplicity. However, because there is only one
order3 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 order3 for the sample of the 8 aspects of a magic
square. Of course, normalizing order3 is meaningless, because there is only one
square of this order. The 440 normalized (or basic) magic squares of order4
were published in 1693 (Frénicle 16051675), so such a list has real meaning for
this, and higher orders. [1][2]
Interestingly, the number of order5 magic squares was not
established until December, 1973. At that time, Richard Schroeppel calculated
and proved that there were 275,305,224 basic order5 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 order3 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 order3 cubes,
but presumably without realizing that that was all there were.
[6]
J. R. Hendricks proved in January 1972 that only four order3 basic magic cubes
were possible. [7].
Hendricks found all 58 basic order3 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 order4. 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
order4 magic squares is available for downloading from my
downloads page. 

Text list of the first 4 order4 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:
 Put
the lowest corner number in the lower left front corner.
 Then
assign the adjacent numbers in increasing magnitude in the x, y, z, …
directions
Here are
the four basic order3 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 reconstructed 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 
Order3 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 order3 magic hypercubes can be reconstructed from a small
group of numbers in a catalogue list. This is because order3 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 3dimensional 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 nonnormalized aspects of index number 57 (of the 58 order3 magic
tesseracts).
The normalized version has13 in the
lower left front corner, with the 4 adjacent numbers 54, 57, 62, 75.
See my
order3 Tesseracts page for a catalogue list of all 58 order3 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,
0486232360
[3] Schroeppel,
Richard, Appendix 5: The Order 5 Magic Squares, 1973, 116
This is a report of Schroeppels work with enumeration of
order 5 magic squares , writeup by M. Beeler, mentioned by Gardner
in his Scientific American column in January 1976.
[4] Walter Trump,
http://www.trump.de/magicsquares/, in an personal email attachment
dated April 8, 2007.
[5] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon:
Illustrated, 2000, 0968798500
[6] W. S. Andrews, Magic Squares & Cubes, 2^{nd}
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, 5:1:1972:4350
[8] Hendricks, John R., Magic Squares to Tesseract by Computer,
1998, 0968470009
Hendricks, John R., All ThirdOrder Magic Tesseracts,
1999, 0968470025
