This is first of 2 pages Of illustrations
of magic squares, cubes, and tesseracts.
This page covers squares
and cubes (2D and 3D), Page 2 is about tesseracts (4D)
What does a magic
hypercube look like?
Magic squares, being 2
dimensions, are easy to illustrate on a 2 dimension piece of paper.
Magic cubes are 3
dimensional, and therefore more difficult to show on paper or a computer screen.
The magic tesseract is the 4
dimensional hypercube. It is much more difficult to construct a meaningful
diagram of this object in two dimensions.
These objects have also been
depicted in other forms. Artwork, models, handicrafts and amulets have all been
used for this purpose.
This, and the next page,
will show examples of how these magic hypercubes have been presented from the
past to the present.
Magic
Squares
The order3 magic square is
the simplest to construct. It’s history goes back to at least the second
millennium BC. It was the subject of much folklore and was called the Luoshu
(The Scroll of the river Luo).
[1]
The first textual reference
to the Luoshu seems to be by ZhuangZi (369286 B.C.E.), However no images are
available from those ancient times.
from page 15 Legacy of the Luoshu diagram by Zheng Xuan (906989) 
from page 11 Legacy of the luoshu

from page 92 Legacy of the Luoshu
This is a 17 century Japanese version 
Some
modern representations of the Luoshu
Ancient legend has it that a tortoise with numbers
inscribed on it's shell visited Sage King Yu, (who died in 2197 B.C.E.) the
founder of the Xia dynasty. Interpretation of this number array was
considered sacred, ritual practice. As a result, the early Chinese had no
interest in investigating magic squares of higher orders.
This magic square is the only one possible for order
3, if rotations and reflections are not considered.
It soon appeared in other early civilizations, and in
virtually all cases was also considered to have magical and mystical powers.
However, these other peoples choose to investigate its features, and device
methods of constructing higher order magic squares.
Other spellings sometimes used are LhoShu, LohShu or
Lo Shu. 
Magic
squares in other cultures
The higher orders of magic
squares started appearing about 1300 A.D.
[2]
Three cultures are known to
have created magic squares, the Chinese, the Indian, and the Arabic. In each
culture they were viewed as having supernatural properties.”
[2]
The first order 4 magic
square seemingly originated in first century in India by a mathematician named
Nagarajuna.
Babylonia, Greece, Egypt
– No record that they knew magic squares prior to the Luoshu, and no
development of higher orders.
India – First mention
of magic squares ca. 550 C.E. It was a number square of order 4 using 2 sets of
the digits 1 to 8. It was pandiagonal magic. (2 3 5 8: 5 8 2 3:
4 1 7 6: 7 6 4 1)
First documented evidence of the order 3 square was ca. 900 C.E.
Jaina order 4 squares have been dated as from the 12th or 13th century
[1 p.85].
Later, squares as large as order 14 were constructed.
Tibet  used the
luoshu for fortune telling and as an occult charm, starting in about the 7th
century. No interest in higher orders.
Japan – The luoshu was
introduced to Japan in the year 970. Unlike the Chinese (who considered the
luoshu sacred) the Japanese started investigating magic squares in earnest.
In 1697 a book was published that showed methods of construction for all orders
from 3 to 30.
Islamic World – The
first recorded involvement with magic squares appears in the writings of Jabir
ibn Hayyan during the period 875 –975. The first set of magic squares was
published in the encyclopedia Rasa’ il about 989.
Possibly the most interesting of these squares was a concentric order 7
(containing also an order 5 and an order 3).
Magic squares (especially the order 3) were also considered by the Muslims to
have religious, meditative and occult significance.
Latin Europe – A book,
originally written in Spanish and the translated to Latin under the Latin title
Picatrix introduced Europe to the Islamic magic squares in 1256. It
described how the orders 3 to 7 related to the sun, moon, and planets. The
emphasis was still on the astrological and occult power of magic squares.
Later, investigators began looking at magic squares more from a recreational
mathematics point of view.
Back to China  First
mention of magic squares greater the 3 in China was in 1275. This because they
considered the Luoshu sacred and so did not investigate other orders.
Larger order MS probably came to china from the Arab world via Arab scholars
starting from about the 11th century.
[1] Much material about this
square is taken from Frank J. Swetz, Legacy of the Luoshu, Open Court. 2002
Other spellings sometimes used are LhoShu, LohShu or Lo Shu.
[2] From Mark Swaney’s Magic Square History site at
http://www.ismaili.net/mirrors/Ikhwan_08/magic_squares.html
Now back
to the illustrations
(with a minimum of text)

Enlarged
view of the magic square. This
Albrecht Dürer
[1] engraving released in 1514, probably did more then
any other single event, to popularize magic squares in Latin Europe.
This was probably the first magic square seen in Europe.
Albrecht Dürer was a renowned and gifted painter and mathematician. He had
refined his education by traveling widely.
[1] Albrecht
Dürer,
Melencolia I,
1514 engraving
9 3/8" x 7 3/8" 
About 1315 the Greek schooler, Manual Moschopoulos,
wrote a treatise on the construction of magic squares. This is the first
known mention of magic squares in Europe. However, his work received
little or no notice for over 200 years.
An early Arabic magic square
http://membres.lycos.fr/fusionbfr/JHM/CM/CM1.html
H. C. Agrippa von Nettescheim, De occulta philosophia libri
tres, 1531
1651 English translation at
http://archive.lib.msu.edu/AFS/dmc/arts/public/all/threebooksoccult/ANL.pdf 
Girolamo
Cardano, Practica arithmetice et mensurandi singularis,
(Magic squares for the heavenly bodies), 1539
Note that the order the squares
relate to the heavenly bodies is the reverse of Agrippa's order. The
magic squares themselves are the same.
The order 6 square had special significance for the early Christians
because the total of all 36 numbers in the square is 666.
Below are thumbnails of the other 5 heavenly
body magic squares. Each was also shown using Hebrew characters (not
numbers). 

By the beginning of the sixteenth century, magic squares were beginning to
appear in Western Europe. In 1531, Agrippa's treatise explained his beliefs
in the occult significance of magic squares, and their relationship to the
heavenly bodies. 8 years later, Cardano published a paper in which he showed
the order of the squares reversed from that of Agrippa.
During the rest of the sixteenth and all of the seventeenth centuries
medallions, amulets, and coins were all the rage in western Europe. To the
left is the thumbnail of an image from an early eighteenth century
newspaper.
Below are thumbnails of a series of 7 coins which each show one of the magic
squares from 3 to 9. I do not show the reverse of each coin.
These images were kindly supplied to me
by Paul Heimbach, a German artist who has worked extensively with magic
squares. His website is at
http://www.artype.de/quadrate/index.html

Ozanam (16401717) and Euler (17071783)


Jacques Ozanam
16401717 Récréations
mathématiques et physiques (1694).
Dr. Hutton’s translation of Montucla’s
Edition of Ozanam Edited by Riddle in 1844.

Some modern
alternative representations of magic squares 
1. is an order 5 magic square constructed using dowels
and metal washers to represent the numbers. Suspending the model from the
center demonstrates that it is in balance.
2.is an an order 3 square constructed with needlework
(crossstitch).
3.is a model of an order 4 magic square composed of
dowels and wood blocks. (These three all constructed by myself).
4. is one possible presentation of an order4 magic
square using dominoes.
Above is a modern sculpture in 3D of an order 3 magic
square (not normal). 
1. 
2. 
3. 
4. 
Magic Cubes
In the
last section we saw that it was simple to depict a magic square onto a piece of
paper (or a computer screen). This is because the square and the paper are both
2 dimensional.
To show a magic cube on a piece of paper is more difficult because the cube is 3
dimensional. In fact, it cannot be done except by introducing distortion. To
introduce the subject, I first show several early methods of depicting an
ordinary cube.


The outline of a cube may
be shown with a schlegel diagram. Here the cube is viewed headon, with the
back face shown smaller. The front and back faces are square (as they should
be), but the top, bottom, and sides are distorted.
A second method is the outline on paper of
what the cube would look like if we viewed it from an angle. Again, the
front and back faces are square, but the top, bottom, and sides are
distorted. 
Of
course, neither method is suitable for illustrating a magic cube, because of the
difficulty of placing the numbers in the diagram. A further complication is the
fact that the ‘cells’ which contain the numbers are themselves 3 dimensional
whereas in the square they are 2 dimensional.
Captions:
1. Par B. Violle, Traité complet des Carrés Magiques, 1837
2. Dr. Theod. Hugel,Das Problem der Magischen Systeme, 1876
3. A. H. Frost, Frost, Descriptions of plates, 1878
4. Frost constructed 1877 consists of 9 vertical glass plates, each with the numbers placed on each side.
This image is supplied courtesy of Christian Boyer's A. H. Frost Biography page
Much more updated information on my cubeupdate6 page
5. A.H. Frost, On the General Properties of Nasik Cubes,1878. An order 4 pantriagonal magic cube.
6. J.A.P Barnard, Theory of Magic Squares and Cubes, 1888, p.266
7. One plane of an order 17 magic cube by Gabriel Arnoux deposited April 17, 1887 in the Académie des Sciences.
8. Fermat's cube (1640). From E. Lucas, L'Arithmetique amusante, 1895, page 226
9. 10. 11. Emile Fourrey, Recreations arithmetiques, 8th edition, Vuibert, 2001. Originally published in 1899.
Three different methods of portraying a magic cube.
12. W.S. Andrews, Magic Squares and Cubes, 1908 (and 1917), p. 65
13. W.S. Andrews, Three orientations of the planes of the previous cube (p.66)
14. H.A, Sayles, General Notes on the Construction of Magic Squares and Cubes With Prime Numbers, The Monist,
vol. 28, January 1918, p. 156 Note the one composite number in his order3 cube example.
15. Ingenieur Weidemann, Zauberquadfrate und andre magische Zahlen figuren der Ebene und des Raumes, 1922, p. 56
(Magic Squares and Other Plane and Solid figures)
16. Max Lehmann, Der geometrische Aufbau Gleichsumiger Zahlenfiguren, 1932, p.285
(Geometric Construction of Magic Figures)
17. R.V. Heath, A Magic Cube With 6n^3 cells, American Mathematical Monthly, vol. 50, 1943, pp 288291
Many of the above cubes are described in more detail elsewhere on my site.


In January 1972, these two illustrations were
published in the Journal of Recreational Mathematics. This was a new
way to illustrate a magic cube.
I have not been able to locate an earlier example of a published
illustration of this type.
This new method may be the result of two earlier papers published by
John Hendricks in The Canadian Mathematical Bulletin and The American
Mathematical Monthly in 1962 and 1968 (see section on
4D). Ironically, those dealt with
depicting a 4dimensional object in two dimensions! 
The numbers now appear at the intersection
of grid lines. Previous illustrations of magic squares and cubes had always
shown the numbers placed in cells between the grid lines. This is now the
preferred method of illustrating the composition of a magic cube. Admittedly,
though, this is really only practical for cube orders up to 5 or 6. For larger
orders, and for occasions when an illustration is not required, the numbers are
usually just presented (normally horizontal) plane by plane.
The above order4 cube in text form
01 32 49 48 62 35 14 19 04 29 52 45 63 34 15 18
56 41 08 25 11 22 59 38 53 44 05 28 10 23 58 39
13 20 61 36 50 47 02 31 16 17 64 33 51 46 03 30
60 37 12 21 07 26 55 42 57 40 09 24 06 27 54 43
top layer second layer third layer bottom layer
Of course, sometimes special circumstances require
special diagrams.
I conclude this section on 3dimensional illustrations with several of an
order8 inlaid magic cube constructed by John Hendricks in 1999.
1. 
2. 
3. 
4. (click on image to enlarge) 
[1] Clifford A. Pickover, The Zen of
Magic Squares, Circles, and Stars, Princeton Univ. Pr., 2002,
2001027848, p. 179
[2] [3] [4] John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd
Edition, Self Published, 1999, 0968470033, pages 155, 158, 166.
Edited and illustrated by Holger Danielsson
A color illustration plus full listing and description of the order8
inlaid magic cube is is at
hendricks.htm.
Now, on to 4dimensional
hypercube illustrations.
