file · Web viewcircles and closed dots on the diagram represents a third order magic square, and is called the . Lo . Shu. The story is legend, but the square is real

T. Glenn Blakney

Professor Reva Kasman

MSM 717

December 10, 2015

How can a square be magic? Its symmetry has an aesthetic appeal, and

the shape is useful for everything from sock drawers to houses to game boards.

But when you create a square table with equal rows and columns of carefully

chosen numbers, that square takes on unusual properties that for some hold

sacred meaning, for others a research tool in Number Theory, and for others a

pleasant diversion on a rainy afternoon. A Magic Square is an array of cells in

three rows and three columns, each containing a non-negative integer, such that

the sum of each column, each row, and both of the main diagonals is the same;

the magic constant or the magic number.1,2 The number of rows or columns is

called the order, or N, and since the numbers are a sequence from 1 to N2, we

can write a formula to calculate the magic constant as: 1+2+3+…+N2

N or more

simply: 12(N3+N )3 The square remains magic under each of eight

transformations: rotations through multiple degrees of 900, including zero, and

the complementary mirror images.

The history of the magic square centers around three ancient cultures:

Chinese, Indian, and Islamic. The earliest story refers to a diagram depicted on

the carapace of a tortoise, which crawled out of the Lo River and presented itself

to a Chinese emperor of the Shang Dynasty of 2300 B.C.E. The series of open

circles and closed dots on the diagram represents a third order magic square,

and is called the Lo Shu. The story is legend, but the square is real.

The earliest actual documentation of magic squares comes from sources in India

and China and dates the discovery and serious study from the first century

B.C.E. in China and the first century C.E. in India. Chinese practitioners looked

to the orientation of the numbers to represent competing cosmological forces and

the natural order of the universe, as represented by the concepts of Yin and

Yang. Odd numbers were Yang and even numbers were Yin; the orientation of

the Lo Shu places these in balance.5 The product of the central number, 5, and

the order of the Lo Shu, 3, equals the sum of each row, column and both

diagonals. Five is also the arithmetic mean of the numbers above and below it,

as well as to the right and left, and the corners. The square of the order,

multiplied by the center number yields the sum of all the elements in the magic

square. All odd-numbered magic squares follow these rules.6 It was these

properties that captivated Chinese mathematicians and fortune tellers for about

T. Glenn Blakney MSM717 Magic Squares December 10, 2015

2

800 years until interest faded and the agencies that sustained the Lo Shu came

into disfavor.7

The study of magic squares in India is dated to the 2nd Century C.E. and is

attributed to the Buddhist philosopher, Nagarjuna.8,9 Indian mathematicians

expanded the exploration of squares by constructing 4th Order squares and

devising rules to construct squares with odd and even totals.10 Starting with a

series of numbers suggested by Nagarjuna, (inserted in the Indian way—

horizontal rows beginning in the upper left corner)11, a foundation of elements is

set and the empty cells are filled with an expression which is the difference of

half the constant and its complementary cell.

In 1356 C.E., Narayana, the Indian mathematician, lays out a detailed study of

magic squares which includes specific methods for construction of large and

complex squares, as well as theorems which state their properties.12

Hard evidence of the deliberate construction of magic squares in the

Islamic world dates to roughly 900 C.E.13 Using the Lo Shu and what rules for

construction that they could deduce, Muslim mathematicians quickly began to


3

explore the construction of 4X4 and larger magic squares.14 Because they did

not invest the squares with religious value, they were free to sample from both

the Chinese methods and the Indian methods, so they tried combining squares,

using different numbers, swapping diagonals, columns and rows and trying

numbers in different series and adding borders. As their skills developed, they

used the Lo Shu as a core and built squares around it, or they used those rules

to start with a central cross and worked from there.15 Around the 13th Century,

Islamic mathematicians began to perceive the orders in a square as the flow of a

continuous process. If they rotated a Natural Square (integers in ordered rows,

starting in one corner or another) 450 to the right, some of the numbers fell

outside the square. Those numbers now filled the empty squares on the opposite

side of the square, in this manner:

Rolling the paper into a cylinder, matching the top and bottom, or the right and

left sides, and the numbers spiral in sequence; this is the continuous system for

numbering squares, a uniquely Islamic method.16


4

Classifying Magic squares is analogous to classifying the organisms that

occur in nature: new patterns, constructions, and properties are discovered on a

regular basis, and need to be studied and grouped.17 As we have noted, a simple

Magic Square is one where the sum of both diagonals, each column, and each

row and each row is a constant. The Lo Shu is a Simple Magic Square. The

unique property of Associated Squares is that the sum of any two numbers

located in cells diametrically equidistant from the center of the square equals the

sum of the first and last terms of the series, or the square of the order, N, plus

one.18 There is a class of squares which retain their magic properties when every

element is squared, or cubed, or even raised to its n th power. (Pickover is

unaware of a 4th or 5th power magic square)19. Nasik, Diabolic, or Pandiagonal

magic squares have a unique property in addition to those of simple squares: all

of their broken diagonals add to the magic constant. (A broken diagonal is a

constant series of numbers that extends off the edge of the square and

reappears at the opposite side). Magic squares can even be made of any

arithmetic series such as odd or even numbers, by making each term twice the

previous, or even by employing some creative operation involving the Order.

These are called Imperfect magic squares.20 Not all the squares in every class

have been discovered, so there are many opportunities for exploration and

discovery.

Starting just from the Lo Shu, it is simple to begin an exploration of magic

squares. For instance, if all the elements maintain their initial relative positioning,

and each element is multiplied by any real number; all of the original properties of


5

the magic square remain! Add 9 to each element to make a new square. Repeat

this eight times and label each one, one through nine. Now arrange these in a

grid, each numbered square placed where its number appears in the Lo Shu.

You may rotate them or reflect them in any way. You have just created a 9X9

magic square! In fact, using the 81 original integers, you can create over one

hundred million ninth order magic squares!21 Working off established rules and

properties which others have already discovered, the real fun of magic squares is

the opportunity for create new squares and to reveal new relationships in old

squares.


6

Citations

1. Andrews, W. S., and L. S. Frierson. Magic Squares and Cubes. (New York:

Dover Publications, 1960) pp. 1

2. Gardner, Martin. "Seventeen." Time Travel and Other Mathematical

Bewilderments. (New York: W.H. Freeman, 1988. 213-19) pp. 214

3. Ibid. 214

4. Swetz, Frank. Mysticism and Magic in the Number Squares of Old China (The

Mathematics Teacher 71.1 1978): pp. 51

5. Ibid. pp. 51

6. Ibid. pp. 51

7. Cammann, Schuyler. "The Magic Square of Three in Old Chinese Philosophy and

Religion." (History of Religions 1.1 1961) pp. 76

8. Datta, Bibhutibhusan, and Singh, Awadhesh Narayan. "Magic Squares in India."

(Indian Journal of History of Science 27.1 1992) page 590

9. Sridharan, Raja and Srinivas, M.D. “Folding Method of Narayana Pandita for the

Construction of Samagarbha and Visama Magic Squares” (Indian Journal of

History of Science, 47.4 2012)

10. Datta, Singh 52

11. Cammann, Schuyler. “Islamic and Indian Magic Squares. Part I, II”. History of

Religions 8.4 (1969): 274. Web...

12. Datta, Singh 59

13. Cammann 190

14. Ibid. pp. 190

15. Ibid. pp. 193

16. Ibid. pp. 198

17. Pickover, Clifford A. The Zen of Magic Squares, Circles, and Stars: An

Exhibition of Surprising Structures across Dimensions. (Princeton, NJ: Princeton

UP, 2002) pp. 38-56

18. Ibid. pp. 65

19. Ibid. pp. 68

20. Ibid. pp. 84


7

21. Trigg, Charles W. "A Family of Ninth Order Magic Squares." (Mathematics

Magazine 53.2 1980): pp. 100


8

Bibliography

Andrews, W. S., and L. S. Frierson. Magic Squares and Cubes. New York: Dover

Publications, 1960. Print.

Boyer, Carl B. A History of Mathematics. New York: Wiley, 1968. 197+. Print.

Cammann, Schuyler. "The Evolution of Magic Squares in China." Journal of the

American Oriental Society 80.2 (1960): 116-18. Web.

Cammann, Schuyler. "Islamic and Indian Magic Squares. Part I, Part II." History of

Religions 8.3 (1969): 181-209, 271-299. Web.

Cammann, Schuyler. "The Magic Square of Three in Old Chinese Philosophy and

Religion." History of Religions 1.1 (1961): 37-80. Web.

Carus, Paul. "Reflections on Magic Squares: Mathematical, Historical and

Philosophical." Monist 16.1 (1906): 123-47. Web.

Datta, Bibhutibhusan, and Awadhesh Narayan Singh. "Magic Squares in India." Indian

Journal of History of Science 27.1 (1992): 51-120. Web. 1 Dec. 2015.

<http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005b5a_51.pdf>.

Gardner, Martin. "21: The Return of Dr. Matrix." Penrose Tiles to Trapdoor Ciphers...:

And the Return of Dr. Matrix. Washington, DC: Math. Assoc. of America, 1997.

293-305. Print.

Gardner, Martin. “Some New Discoveries About 3 × 3 Magic Squares”. Math Horizons

5.3 (1998): 11–13. Web...

Gardner, Martin. "Seventeen." Time Travel and Other Mathematical Bewilderments.

New York: W.H. Freeman, 1988. 213-19. Print.


9

"Heilbrunn Timeline of Art History." Han Dynasty (206 B.C.–220 A.D.). Department of

Asian Art, The Metropolitan Museum of Art, 2000, n.d. Web. 05 Dec. 2015.

<http://www.metmuseum.org/toah/hd/hand/hd_hand.htm>.

Holzman, D. "Bibliographie." T'oung Pao 55.4 (1969): 317-44. Web.

Hummel, Siegbert. “The Sme-ba-dgu, the Magic Square of the Tibetans”. East and

West 19.1/2 (1969): 139–146. Web...

Pickover, Clifford A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of

Surprising Structures across Dimensions. Princeton, NJ: Princeton UP, 2002.

Print.

Swetz, Frank. "Mysticism and Magic in the Number Squares of Old China” The

Mathematics Teacher 71.1 (1978): 50-56. JSTOR. Web. 05 Dec. 2015.

Sridharan, Raja and Srinivas, M.D. “Folding Method of Narayana Pandita for the

Construction of Samagarbha and Visama Magic Squares” Indian Journal of

History of Science, 47.4 (2012) 589-605

Trigg, Charles W. "A Family of Ninth Order Magic Squares." Mathematics Magazine

53.2 (1980): 100. Web.

Weisstein, Eric W. "Magic Square." Mathworld/Wolfram. Wolfram, n.d. Web. 12 Dec.

2105. <http://mathworld.wolfram.com/magicsquare.html>.


10

Documents

file · Web viewcircles and closed dots on the diagram represents a third order magic square, and is called the . Lo . Shu. The story is legend, but the square is real