WOMEN IN MATHEMATICS MARGARITA PANAYOTOVA Melancholia Albrecht Drer

WOMEN IN MATHEMATICS MARGARITA PANAYOTOVA

Melancholia, Albrecht Dürer, 1514 Engraving, 31 X 26 cm

Albrecht Dürer (1471 -1528) l Self –Portrait at 28 (Self-Portrait in Fur Coat)

Albrecht Dürer l Albrecht Dürer was a German painter, printmaker, mathematician, and, with Rembrandt and Goya, the greatest creator of old master prints. l The foundations of descriptive geometry are laid in Dürer's treatise on human proportions published in Nuremberg after his death in 1528.

Melancholia Web. Museum, Paris. l An allegorical composition which has been the subject of very many interpretations. l

l Albrecht Dürer’s Magic Square, 1514

Magic Squares Magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. l A normal magic square contains the integers from 1 to n². l

Sagrada Familia, Barcelona

The Sagrada Familia Magic Square 1 14 14 4 11 7 6 9 8 10 10 5 13 2 3 15 The magic constant of the square is 33, the age of Jesus at the time of the Passion. l While having the same pattern of summation, this is not a normal magic square, as two numbers (10 and 14) are duplicated and two (12 and 16) are absent, failing the 1→n² rule. l

l How many magic squares with dimensions m x m are there? l How can we construct them?

l The On-line Encyclopedia of Integer Sequences gives the sequence: 1, 0, 1, 880, 275305224, … l The 17 th century amateur mathematician, Bernard Frénicle de Bessey, determined that there are exactly 880 essentially different 4 x 4 magic squares, i. e. squares that cannot be obtained from one another by rotations or reflections. He did this by an exhaustive search, listing all 880 possibilities. l In 1973 Richard Schroeppel computed the 5 x 5 magic squares.

Generic Pattern for Construction l All the numbers are written in order from left to right across each row in turn, starting from the top left hand corner. Numbers are then either retained in the same place or interchanged with their diametrically opposite numbers in a certain regular pattern.

Construction of a Magic Square of Order 4 (1) Draw a 4 x 4 square and go through the boxes one row at a time, left to right, top to bottom, counting from 1 to 16, but writing down the number of the box only when it falls on the diagonal.

Construction of a Magic Square of Order 4 (2) Then count down from 16 to 1, and using only the numbers not yet in the square, fill in the boxes that are left (see numbers in red).

Notice that the four entries in the upper left-hand corner of the left square (1, 15, 12, 6) add up to 34, which is the same as the sum for each row, column and main diagonal, and that the four in the upper right-hand corner (14, 4, 7, 9), the four in the lower left (8, 10, 13, 3), and the four in the lower right (11, 5, 2, 16) also add up to 34.

l The magic squares problem leads to work into areas of mathematics such as theories of groups, lattices, Latin squares, determinants, partitions, matrices, and congruence arithmetic. l The magic squares problem is an example of a problem which is easy to understand, yet hard to solve in general. In fact, the problem of determining how many magic squares there are of size n has not been solved.

Magic Squares and Dither Printing l About 1990 an electrical engineer who was interested in the use of the pandiagonal magic square in the process of dither printing used for the fast production of pictures in newspapers sent a letter to a woman. l This letter motivated the woman to do research of pandiagonal magic squares of order n, where n is a multiple of 4(no pandiagonal magic squares exist of order n=4 k+2).

l Pandiagonal Magic Squares A pandiagonal magic square, in addition to satisfying the requirement of a magic square, has the additional property that all diagonals, including broken diagonals, i. e. those that wrap around from one edge of the square to the opposite edge, add to the same sum. 0 62 2 60 11 53 9 55 15 49 13 51 4 58 6 56 16 46 18 44 27 37 25 39 31 33 29 35 20 42 22 40 52 10 54 8 63 1 61 3 59 5 57 7 48 14 50 12 36 26 38 24 47 17 45 19 43 21 41 23 32 30 34 28

Most-Perfect Pandiagonal Magic Squares: Their Construction and Enumeration The woman worked on most-perfect pandiagonal magic squares for over eight years. In a 1986 paper she made use of symmetries to determine that there are 368, 340 essentially different such squares of order 8. l Slowly she figured out how to construct and how to count the total number of squares, first for those whose order is a power of 2, then for squares who order is a multiple of a power of 2, and finally, after another four years of work, for all most-perfect pandiagonal magic squares with order a multiple of 4. l

l The book was published in 1998 to international acclaim and provided for the first time an algorithm for constructing a whole class of magic squares as well as a formula for counting their number, a remarkable accomplishment for a woman of age 85.

Dame Kathleen Timpson Ollerenshaw l Kathleen Timpson was born in Manchester, England on October 1, 1912. She is the younger of the two daughters of Charles and Mary Timpson. Her passion for numbers began as a young child.

l In 1921, when she was 8 years old, the combination of a viral infection and family genetic history led to a sensori-neural deafness. l She quickly learned to lip-read, a skill that allowed her to succeed in school and university, and at her first job. l At the age of 37 she received her first effective hearing aid. l Throughout her life, however, she has never allowed this handicap to restrict her activities.

l Ollerenshaw attended the Ladybarn House School until the age of 13. l At this school, also, at the age of 6, she first met Robert Ollerenshaw who would later become her husband. l Her passion for numbers didn’t seem to come from anywhere. Neither mother nor father showed any special interest. l The Headmistress of the school was a mathematician. She took a special interest in Kathleen, emphasizing to her the need for proof and the difference between conjecture and logical mathematical arguments.

l “Mathematics is the one school subject not dependent on hearing. I was lucky with my teachers, but I was also to a large extent selftaught; reading books about the great mathematicians, solving problems set in magazines. ” l After finishing school in 1930, Ollerenshaw entered Somerville College at Oxford to study mathematics, and graduated from it in 1934. l An excellent athlete, she played hockey, and served as team captain. l In Oxford she met again Robert, who was studying physiology in preparation for a medical career.

l In 1936 she began working in the statistics at Liaison departments of Shirley Institute, a cotton research establishment. l She continued to play hockey for various local and regional teams, and competed in figure skating competitions (in February 1939 she was runner-up in the English-style British Pairs Ice-Skating Championship).

l Kathleen and Robert were married in September 1939, and Robert left almost immediately to serve with the British medical corps in World War II, although he remained in England until sent to North Africa in 1942.

l Through the head of the mathematics department at Manchester University, Ollerenshaw came to know Kurt Mahler, a mathematician who had come to Manchester from Germany in 1938.

l Mahler Lattice Points in a Circular Quadrilateral Bounded by the Arcs of Four Circles Quarterly Journal of Mathematics, Oxford Series 17, (1946) 93 -98 [Received 1 May 1945] mentioned to Kathleen an unsolved problem on critical lattices, an area combining aspects of number theory and geometry, which she solved within a few days. Impressed with her abilities, Mahler suggested she consider returning to Oxford for a DPhil degree.

l In 1943 Kathleen returned to Somerville College. l Over the next two years, while caring for her son, while her husband was away at war, she wrote five original research papers which were sufficient for her to earn her DPhil degree without the need of a formal written thesis. She received her degree in 1945 just as the war was ending and shortly before Robert returned to England.

l In 1946 Kathleen gave a birth to a daughter, Florence. l The year 1953, however, saw the beginning of a life-long involvement in politics and educational issues in England Wales. l In 1954 she was appointed to the Manchester Education Committee. l Two years later she won election to the Manchester City Council as a member of the Conservative party, a position she held for the next 25 years.

l From 1958 to 1967, Ollerenshaw was chairman of the Association of Governing Bodies of Girls' Public Schools. l She wrote two books and many articles in defense of schools for girls and girls' education in general. l She has served on the governing bodies of five universities in the northwest region of England, including the Royal Northern College of Music which she helped to establish in the early 1970's.

USSR l Ollerenshaw's interest and expertise in educational issues led to several trips abroad. l In 1963, as a member of a delegation from the British Association for Commercial and Industrial Education, she spent three weeks in Russia visiting technical colleges, schools, and universities to learn about Russian postschool vocational education and training.

USA, JAPAN Two years later she received a Winifred Cullis Lecture Fellowship to visit the United States for a three month tour emphasizing mathematics education. l In 1970 the British Council sponsored a visit to Japan to have Ollerenshaw talk about the relationship between local and central governments and to lecture about mathematics education in England Wales. l

Dame Kathleen Ollerenshaw l In 1970 Ollerenshaw was made a Dame Commander of the Order of the British Empire for "services to education" (becoming a dame is the female equivalent of the knighthood).

Knighted Mathematicians l Isaac Newton, knighted in 1705. Newton was the most influential mathematician/scientist in history (and the first to be knighted). l In 1843 William Rowan Hamilton discovered the quaternions, the first noncommutative algebra to be studied. Mathematician, knighted in 1835.

Michael Francis Atiyah was knighted in 1983. l There isn't any outsider who penetrated so deep in population genetics as astronomer- mathematician Sir Fred Hoyle, with such a mathematical knowledge, with such an integrity to find out the truth, and without distorting his subject of investigation. Fred Hoyle was knighted in 1972.

l British Mathematician Sir Roger Penrose. In 1994 Penrose was knighted for his services to science. Sir Erik Christopher Zeeman is one of the great XXth century mathematicians. He was also knighted in 1991.

Professor Wiles (who proved Fermat's Last Theorem) was made a Knight of the British Empire in 2000. Professor John Ball, a worldrenowned mathematician and Sussex alumnus, has received a knighthood in the New Year Honours list for 2006. Now working at Oxford University, he is president of the prestigious International Mathematical Union (IMU).

l Despite the many demands of all her public service, Ollerenshaw did not abandon her own mathematical interests. l A paper that Ollerenshaw published in 1980 in the Bulletin of the Institute of Mathematics and Its Applications gave one of the first general methods for solving the Rubik cube puzzle (or the Hungarian magic cube as it was often called then) that tried to minimize the total number of moves needed. l Ollerenshaw's interest in the mathematical theory of magic squares also began in the early 1980 s.

Ollerenshaw and Sir Hermann Bondi Ollerenshaw and Hermann Bondi, a prominent cosmologist and mathematician, developed an analytical construction of the 4 x 4 magic squares, thereby verifying their number 880. l They published their results in a 1982 paper in the Philosophical Transactions of the Royal Society, which was reprinted a year later as a book. l

Ollerenshaw and David S. Brée l After this success, Ollerenshaw began the study of pandiagonal magic squares of order n. “Most-Perfect Pandiagonal Magic Squares: Their Constructions and Enumeration” was published in 1998.

Lancaster University, Department of Physics l An observatory is situated on the roof of the Lancaster University Physics Department, a department with an outstanding reputation for both teaching and research grade. The observatory is dedicated to the teaching of observational astronomy to undergraduate students as part of their degree scheme in Physics, Astrophysics and Cosmology.

The Dame Kathleen Ollerenshaw Observatory l The observatory, comprising the telescope, dome and the adjacent laboratory, is named after Dame Kathleen Ollerenshaw. Dame Kathleen generously donated the telescope. The observatory was formally opened by Sir Patrick Moore on 20 th May 2002.

To Talk of Many Things Manchester University Press, 2004 l "To Talk of Many Things" is a remarkable account of a remarkable life. This story covers two world wars and the near sixty years that followed in a life dominated by mathematics and public service. l

Dame Kathleen Ollerenshaw “Long developed interests have to be worked on, nurtured and constantly refreshed during the good times. ”