__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 are 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 truly 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 descent 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.

**Written by Bibi Drum, Joy Winstead February 20, 2001**

**http://www.cs.appstate.edu/~sjg/wmm/student/muhammed/Muhammad.htm**