Dame Kathleen Ollerenshaw: Puzzles, bubbles and lattices

Dame Kathleen OllerenshawOriginally published in the ebook A Passion for Science: Stories of Discovery and Invention.

by Katie Steckles

Dame Kathleen Ollerenshaw is a prolific mathematician and political figure originally from Manchester. Even among mathematicians, Kathleen Ollerenshaw isn’t a household name, but she should be: she’s made contributions to several areas of mathematics, and her work in politics included a long campaign to improve the state of education, in particular maths education, in Britain.

Born in Withington in 1912, Kathleen studied at St Leonard’s boarding school, St Andrews, where she excelled in mathematics as well as enjoying sports. Although she lost her hearing at the age of eight due to an inherited condition, she didn’t let this affect her work and studies — she could lip-read fluently. Kathleen considered mathematics to be one of only a few subjects in which her deafness didn’t put her at a disadvantage. She didn’t even reveal to her interviewers at Oxford she was deaf until she’d been accepted as a student.

Having earned her BA at Oxford, Ollerenshaw also completed her doctorate there in 1945 on the subject of critical lattices. Lattices are sets of points arranged in straight lines with the same distance between each pair of adjacent points. They can be two- or three-dimensional, or occur in higher dimensions, and studying them can reveal insights into problems such as finding the best way to arrange objects to optimise the use of space — like arranging tins in a box, or oranges in a crate. These types of questions are called ‘close-packing problems’. It’s possible to consider several different ways of arranging, for example, spheres in three-dimensional space, and decide which gives the most efficient packing — the one with the least space left unused — by placing the centre of a sphere at each point in a lattice.

Rather than publishing a formal written thesis, Ollerenshaw was awarded her DPhil on the basis of five research papers that she published on the topic. These papers discussed critical lattices — ones for which every point in the lattice is the centre of a cube, when the whole space is tiled with cubes (or in two dimensions, squares) in some arrangement. She proved many facts about the nature of these lattices when considered in different spaces and in higher dimensions.

Bubbles and puzzles

After the Second World War, Kathleen and her husband, Robert Ollerenshaw, moved to Manchester where Kathleen worked part-time as a lecturer in the University’s mathematics department. She was invited to become a Founder Fellow of the Institute of Mathematics and its Applications (IMA) in 1964. In 1970 she became a member of the IMA’s governing council and was its president from 1978-1979. Many of her subsequent mathematical discoveries were published in the IMA’s monthly bulletin, now called Mathematics Today.

In her IMA presidential address, Kathleen discussed the mathematics of soap bubbles. Since the liquid which makes a bubble will always arrange itself to have minimal surface area, a floating bubble will be in the shape of a sphere. If a bubble is touching another bubble or object its shape might be different but it will still conform to the minimal area. For example, when two bubbles of the same size are stuck together, they will have a flat face between them, whereas different sized bubbles will have a curved surface which bulges into the larger bubble because the air pressure inside the smaller bubble will be higher.

The behaviour of bubbles can be modelled using mathematical equations, and the resulting geometry of shapes has been used by engineers to optimise the construction of roof structures. It’s also possible to use mathematics to study the way three or four bubbles meet at a point — the walls between them form angles of 120 degrees and 109 degrees respectively — allowing you to to model the way foam forms, which is very useful in the study of such materials.

As well as working on serious mathematical problems, Dame Kathleen also contributed greatly to recreational mathematics — the study of interesting mathematical curiosities and puzzles. In 1982, she published a paper with Hermann Bondi, entitled Magic squares of order four. A magic square is an arrangement of numbers in a grid such that every row and column of the grid, and the diagonals, contain numbers which sum to the same total — the ‘magic constant’. In their paper, Ollerenshaw and Bondi proved a statement made in 1693 by French mathematician Bernard Frénicle de Bessy. The statement was that if you arrange the numbers 1 to 16 in a four-by-four square, there are 880 essentially different ways, ignoring reflections and rotations, to do this which result in a magic square.

Dame Kathleen continued to study magic squares in her spare time for eight years, and looked at the particular class called ‘pandiagonal magic squares’. These have the additional property that broken diagonals, that is the diagonals that wrap round at the edges of the square, also add up to the magic constant. She produced a method for constructing and listing these types of squares, published as part of an IMA bulletin in 1989 and later released in the form of a book, Most Perfect Pandiagonal Magic Squares: Their Construction and Enumeration, in collaboration with David Bree, published in 1998.

She was also among the first people to study and write down algorithms for solving the Rubik’s cube. Her solution algorithm was published in an issue of the IMA Bulletin in 1980; her method started by solving the bottom face first, then the top corners, then the middle slice edges, and finally the top edges. This resulted in a solve which took an average of 80 moves – not nearly as efficient as a competitive ‘speed-solver’ would achieve, but her method would work starting from any of the roughly 42 billion billion possible scrambled cube positions, when followed step-by-step.

Many common solving methods today use the same layer-by-layer approach, including the method given in the leaflet provided with modern cubes. Of course, when it was originally sold, the cube didn’t include any such instructions and Dame Ollerenshaw had to work it out for herself the hard way. She famously injured the tendons in her thumb from working with the cube so much and needed minor surgery — the Reader’s Digest noted this as the first recorded case of ‘mathematician’s thumb’.

Ollerenshaw was generally regarded as a superb mathematician and thinker. She described a method for problem solving which she developed while at boarding school: she would think about a mathematical problem in the moments before going to sleep and trace the shapes and formulae on the wall with her finger. In the morning when she woke, the answers would be there.

Improving maths education

Having served as a school governor, Kathleen developed an interest in the conditions of British schools. She published an article on the topic outlining the declining state of school buildings in the Manchester area, and became a member of various education committees and advisory groups. Later she was elected as a member of Manchester City Council, where she was a councillor from 1956 until 1981 and served on the finance committee. She was also elected Lord Mayor of Manchester in 1975-6.

Kathleen’s interest in improving the state of education, and in particular mathematics education, led her to visit other countries to learn about their systems. She travelled to the USSR to find out about their higher education, and she also visited schools and educators in the USA while working there on a fellowship.

In 1969, researchers at Stanford released the results of a project aiming to measure the standards of mathematics teaching in countries across the world. The results showed that children in Japan were far ahead of those in other countries, so Kathleen obtained sponsorship from the British Council to visit Japan and observe their practices in mathematics teaching. While the class sizes there were larger, she saw a much higher standard of discipline, and their attitude towards mathematics was much more positive.

In 1972, Dame Kathleen became a part-time senior research fellow in the Department of Educational Research at Lancaster University. Her drive to improve standards in education led to her appointment as an advisor on educational matters to Margaret Thatcher’s government in the 1980s.

Dame Kathleen had many other interests — she enjoyed skating and skiing, and was a keen amateur astronomer. She was made an honorary member of the Manchester Astronomical Society, one of the oldest provincial astronomical societies in England. She donated an 11-inch Celestron telescope to Lancaster University, and the observatory there bears her name. She also donated a prize to the University’s physics department, awarded to the best fourth year MPhys project.

At the age of 37, Kathleen was fitted with her first hearing aid, and for the first time in her life was able to hear the sounds of the world around her. In 1968, now in her mid-50s, having been fitted with a more advanced hearing aid, she developed a love of music and was instrumental in the establishment of what would later become the Royal Northern College of Music, chairing its governing body for several years.

Dame Kathleen’s contribution both to politics and mathematics have been well recognised. She was appointed Dame Commander of the Order of the British Empire for services to education in 1970, as well as being awarded Freedom of the City of Manchester for her long service to the city. She holds honorary degrees from the Victoria University of Manchester, the University of Lancaster, and Liverpool University. The mathematics department at the University of Manchester, where she worked as a lecturer, holds an annual Ollerenshaw lecture, given by a visiting speaker, and Dame Kathleen herself has attended the lecture on several occasions. She has published 26 mathematical papers and contributed greatly to the subject. In 2012 she celebrated her 100th birthday, and continues to be an inspiration to mathematicians and educators alike.

About the author

Katie Steckles is a mathematician, based in Manchester, who works in public engagement and lectures at Sheffield Hallam University. She visits schools as part of Think Maths to give talks and workshops, and works at science festivals and other events to promote mathematics. She also writes at The Aperiodical, an online maths magazine blog of news, features and editorials, and has appeared on Channel 4, BBC Radio 5 Live, BBC Teach, BBC Radio Manchester and the Discovery Science Channel.

Posted in STEM Stories.