WOMEN IN
MATHEMATICS
MARGARITA PANAYOTOVA
Melancholia,
Albrecht Dürer,
1514
Engraving,
31 X 26 cm
Albrecht Dürer (1471-1528)
Self –Portrait at
28 (Self-Portrait in
Fur Coat)

Albrecht Dürer
Albrecht Dürer was a German painter,
printmaker, mathematician, and, with
Rembrandt and Goya, the greatest
creator of old master prints.
 The foundations of descriptive geometry
are laid in Dürer's treatise on human
proportions published in Nuremberg
after his death in 1528.

Melancholia

WebMuseum,
Paris.
 An allegorical
composition which
has been the
subject of very
many
interpretations.

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.
A normal magic square
contains the integers from
1 to n².
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.
 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.
 How
many magic squares with
dimensions m x m are there?
 How can we construct them?
The On-line Encyclopedia of Integer
Sequences gives the sequence:
1, 0, 1, 880, 275305224,…
 The 17th century amateur mathematician,
Bernard Frénicle de Bessey, determined that
there are exactly 880 essentially different 4x4
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.
 In 1973 Richard Schroeppel computed the
5x5 magic squares.

Generic Pattern for
Construction

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 4x4 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.
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.
 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
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.
 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=4k+2).


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.
 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.
 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

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.
In 1921, when she was 8 years old, the
combination of a viral infection and family
genetic history led to a sensori-neural
deafness.
 She quickly learned to lip-read, a skill that
allowed her to succeed in school and
university, and at her first job.
 At the age of 37 she received her first
effective hearing aid.
 Throughout her life, however, she has never
allowed this handicap to restrict her
activities.

Ollerenshaw attended the Ladybarn House
School until the age of 13.
 At this school, also, at the age of 6, she first
met Robert Ollerenshaw who would later
become her husband.
 Her passion for numbers didn’t seem to come
from anywhere. Neither mother nor father
showed any special interest.
 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.

“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.”
 After finishing school in 1930, Ollerenshaw
entered Somerville College at Oxford to study
mathematics, and graduated from it in 1934.
 An excellent athlete, she played hockey, and
served as team captain.
 In Oxford she met again Robert, who was
studying physiology in preparation for a
medical career.

 In
1936 she began working in the
statistics at Liaison departments of
Shirley Institute, a cotton research
establishment.
 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).
 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.

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.

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]
Mahler 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.
In 1943 Kathleen
returned to Somerville
College.
 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.
In 1946 Kathleen gave a birth to a
daughter, Florence.
 The year 1953, however, saw the
beginning of a life-long involvement in
politics and educational issues in
England and Wales.
 In 1954 she was appointed to the
Manchester Education Committee.
 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.

From 1958 to 1967, Ollerenshaw was
chairman of the Association of Governing
Bodies of Girls' Public Schools.
 She wrote two books and many articles in
defense of schools for girls and girls'
education in general.
 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
Ollerenshaw's interest and expertise in
educational issues led to several trips abroad.
 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.
 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 and
Wales.
Dame Kathleen Ollerenshaw

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

Isaac Newton, knighted in
1705. Newton was the most
influential
mathematician/scientist in
history (and the first to be
knighted).

In 1843 William Rowan
Hamilton discovered the
quaternions, the first
noncommutative algebra to
be studied. Mathematician,
knighted in 1835.
90
80
70
60
50
40
30
20
10
0
East
West
North
1st
Qtr
3rd
Qtr
Michael Francis Atiyah
was knighted in 1983.

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.
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).
Despite the many demands of all her public
service, Ollerenshaw did not abandon her
own mathematical interests.
 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.
 Ollerenshaw's interest in the mathematical
theory of magic squares also began in the
early 1980s.

Ollerenshaw and Sir Hermann
Bondi


Ollerenshaw and Hermann
Bondi, a prominent
cosmologist and
mathematician, developed
an analytical construction of
the 4x4 magic squares,
thereby verifying their
number 880.
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.
Ollerenshaw and David S.
Brée

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

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

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 20th May
2002.
To Talk of Many Things
Manchester University Press, 2004


"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.
Dame Kathleen Ollerenshaw
“Long developed
interests have
to be worked
on, nurtured
and constantly
refreshed
during the good
times.”