Bibi Drum

Joy Winstead

February
20, 2001

__Muhammad: a Life in Math, Magic,
and Religion.__

Have you ever wondered how mathematics,
magic, and religion are all connected?
Look no further than the work of Muhammad ibn Muhammad al-Fullani
al-Kishnawi, of Katsina (now Nigeria).
Although not much is known about Muhammad’s life, what we do have are
his quotes and written words that reveal to us what type of person and
mathematician he became. We also
know what type of math Muhammad worked on through the reading of Africa
Counts.

There is still debate as to what year
Muhammad was born, however, we do know that his time was spent creating a new
way to develop magic squares and completing the five pillars of Islam. His multi talents as an astronomer,
mathematician, mystic, and astrologer helped him during his prolific
career. As a member of the Fulani
people, he was one of the first groups to be converted to Islam. The Fulani people have a history as
nomadic herders and traders; they also have made an impact on politics and
economics throughout West Africa.
Additionally, the Fulani people are very independent and
competitive. They have used Islam
as well as their competitive spirit to acquisition new lands around present day
Nigeria.

Because of Muhammad’s faith, he spent a large portion of his
life in the Middle East completing his duties as a devoted Muslim. It is because of this devotion to
Islam, that he is recorded as saying, “work in secret and privacy. The letters are in God’s safekeeping.
God’s power is in his names and his secrets, and if you enter his treasury you
are in God’s privacy, and you should not spread God’s secrets
indiscriminately.” This quote from
Muhammad clearly symbolizes the first pillar of Islam by stating that any
inspiration that is given to you by God, stays between you and God until
another is found worthy of this inspiration. This leads us to the conclusion that Muhammad worked
independently and led his students to do the same. After completing the fifth pillar of Islam, which is the
pilgrimage to Mecca, Muhammad traveled to Egypt. While there, in 1732, he wrote a manuscript in Arabic about
how to complete magic squares of up to an order of eleven. Unfortunately, Muhammad ibn Muhammad
died in Cairo in 1741 before returning to Katsina.

Does
it bother you when you believe you have mastered a concept only to discover you
have not even come close? Do not
worry because some things are not always the way they appear. In the words of Muhammad, “Do not give
up, for that is ignorance and not according to the rules of this art. Those who know the arts of war and
killing cannot imagine the agony and pain of a practitioner of this honorable
science. Like the lover, you
cannot hope to achieve success without infinite perseverance.” This quote describes the pain and
suffering of someone who does not live up to his full potential by giving
up. Muhammad’s statement reveals
to us the quality of his work as a mathematician. He was not only devoted to the art of mathematics, but Muhammad
wanted his students to understand and join him in God’s privacy. This could not be achieved without time
and energy, devotion, and practice.
Without a doubt, giving up is not an option.

Curious as to what this has to do with math, magic,
and religion? The answer goes back
centuries to a divine turtle Lo
Shu in ancient China. On the back
of this divine turtle, appeared this configuration of numbers:

4 |
9 |
2 |

3 |
5 |
7 |

8 |
1 |
6 |

Notice
anything magical about this square?
Look closely and you will find that all rows, all columns, and the two
main diagonals sum to fifteen. This arrangement of numbers in which the
columns, rows, and main diagonals sum to the same number is known as magical
squares. For instance, the row
consisting of four plus nine plus two is equal to the column of four plus three
plus eight, which is equal to the diagonal of two plus five plus eight. All of these sums are equal to
fifteen. The mysterious number
fifteen is known as the magical constant. Muhammad’s work in the mathematical
arts consisted of developing a system to come up with higher order magical
squares. The order of a magic
square is found by counting the number of rows and columns. For example, the magic square that
appeared on the divine turtle Lo Shu, above is of order three. All magic squares have an odd
order. The odd order is necessary
because an even order square does not comply with every property of a magic
square. For example, one can have
an even order in which the columns and rows add to the same. However, the diagonals of the square
will not sum to the same magical constant. The numbers will repeat themselves, and in a true magical
square the numbers are used only once.
The numbers used in a magic square can be found by multiplying the
number of rows by the number of columns.
This is also the same as squaring the order, which is found by counting
the number of rows or columns. For
instance, if there is a three by three magic square, you will use numbers one
through nine.

Muhammad came up with a formula to find
the magical constant, the number that is the sum of the rows, columns, and
diagonals and a formula to find the middle square. The formula for finding the magical constant is n(n^2
+ 1)/2, where n is equal to the order of the magic square. The second formula that Muhammad
developed was

(n^2
+ 1)/2. Once again, n is the order
of the square and in this formula we can derive the middle number.

Muhammad’s
work on magic squares was a beginning to group theory. By group we mean that a set of elements
is closed, associative, contains an identity, and contains inverses for each
element. Muhammad noticed that you
could perform certain operations such as reflection about an axis or rotations
of up to any degree and not change the properties of the square. This meant that out of one simple
square one could now generate a finite number of magic squares and the
properties would still hold true.
For example, the following magic squares are the same square as above
reflected about the x-axis and rotated ninety degrees.

8 |
1 |
6 |

3 |
5 |
7 |

4 |
9 |
2 |

This square is rotated about the

x-axis.

2 |
7 |
6 |

9 |
5 |
1 |

4 |
3 |
8 |

This square is rotated about an

Ninety-degree angle.

Muhammad
proved that combinations of these two reflections are the dihedral group. In other words these two reflections
generate the rest of the group. In this case generate means that all
combinations of these two reflections produce a finite number of elements. There are eight distant elements in
this group. They include the
identity and its inverse and the inverse of every other element. This group is also associative and is
closed under the compositions.
Only the square position is reflected, not the numbers. This is so you do not end-up with an E
for a three.

Although Muhammad ibn Muhammad al-Fullani
al-Kishnawi was not a minority in either race or religion in the western part
of Africa, he was considered a minority because of his career as a
mathematician. Also in the
mathematical world he was one of the few who were not Anglo-Saxon or Christian. Ideas that included people
of African decent could not do mathematical problems, and were intellectually
inferior kept on in the minds of Anglo-Saxons until recently. Despite this Muhammad never once
gave up. He persevered through it
all, never giving in to the pressures of being a minority in both race and
religion. Muhammad showed the
people of the time as well as today that no matter what race, ethnicity, or
religion, you should not let this stand in the way of what you want to do with your life. If Muhammad had let the issues of
multiculturalism get in the way he would have never developed the mathematical
formulas and concepts of group theory that are still used hundreds of years
later.

References

Zaslavsky, Claudia. __Africa Counts: Number and Pattern in
African Culture__.

Boston
Massachusetts: Prindle, Weber & Schmidt, Inc, 1975

Gerdes, Paulus. “On Mathematics in the
History of Sub-Saharan Africa.”

__Historia
Mathematica__. Volumn 21(1994) 345-376

“Muhammad ibn Muhammad al-Fullani
al-Kishnawi” Online.
Internet. 10 Jan.

2001.
Available:
http//www.math.buffalo.edu/mad/special/Muhammad_

Ibn_Muhammad.html.

“Art & Life in Africa” Online. Internet. 15
Feb. 2001. Available: http//www.uio

a.edu/~africart/toc/people/Fulani.html.