post

Routes: Edsger Dijkstra

In a 2001 interview with Philip L Frana for the Charles Babbage Institute, Edsger Dijkstra succinctly demonstrated the importance of his shortest path algorithm:

Dijkstra: If, these days, you want to go from here to there and you have a car with a GPS and a screen, it can give you the shortest way from your hotel to Robbie Creek Cove. Yes?
Frana: I did generate a map, yes, on line.
Dijkstra: Did you use it?
Frana: Yes, I did use it.
Dijkstra: Then you used my algorithm this morning.

The roots of mathematics reach back through time for centuries, providing a solid foundation by anchoring our knowledge in the ages of Pythagoras, Euclid, Fibonacci, Galileo, and Newton. However, some areas of mathematics are new. Their roots are not deep, yet they still carry the huge weight of technological advances that we use, and take for granted, every day.

Continue reading

post

Significant figures: Sir Christopher Zeeman

At the Chalkdust issue 06 launch party, we brought along a challenge for our guests: we connected two people up with ropes and challenged them to separate themselves. To make things interesting, they weren’t allowed to remove the ropes from their hands, cut the ropes or untie the knots. Although the trick is 250 years old, it was made popular by Sir Erik Christopher Zeeman who used it as an interactive way to demonstrate topology, hence the challenge became known as Zeeman’s ropes.

Erik Christopher Zeeman was born in 1925 in Japan to a Danish father, Christian Zeeman, and a British mother, Christine Bushell. A year after his birth they moved to England. Zeeman was educated at Christ’s Hospital, an independent boarding school in Horsham, West Sussex. He did not enjoy the experience, feeling it was a prison in which he lost his self-esteem.

In 1943–1947, Zeeman served in the RAF as a flying officer. In his own words:

“I was a navigator on bombers, trained for the Japanese theatre, but that was cancelled because they dropped the atomic bomb a week before we were due to fly out. Since the death rate was 60% in that theatre it probably saved my life, but at the time I was disappointed not to see action, although relieved not to have to bomb Japan, the land of my birth.”

During his service, Zeeman forgot much of his school maths. But this didn’t stop him from going on to study maths at Christ’s College, Cambridge, where he earned his MA.

Zeeman stayed on in Cambridge for his PhD, in which he wrestled with unknotting spheres in 5 dimensions, spinning knots in 4 dimensions, as well as trying (and failing) to solve the Poincaré conjecture (which would only be resolved in 2005 by Gregori Perelman). He was supervised by Shaun Wylie, who had worked with Alan Turing at Bletchley Park during the war on projects including deciphering a German teleprint cipher called Tunny.

Founding Warwick

In 1963, Zeeman was invited to join the newly established University of Warwick as the foundation professor of mathematics. He initially declined, since he believed Cambridge to be “the centre of the mathematical world”. However, after “a sleepless night”, Zeeman changed his mind and made the biggest move of his life in 1964.

At Warwick, Zeeman was determined to “combine the flexibility of options that are common in most American universities, with the kind of tutorial care to be found in Oxford and Cambridge”. Initially, he recruited lecturers in three main branches of mathematics: analysis, algebra and topology. Legend has it that those he invited to Warwick all declined their offer; his response was to encourage them by telling them that all the others had accepted his invitation. Later on, Zeeman also appointed six lecturers in applied mathematics.

Zeeman building

The Zeeman building, home to the University of Warwick’s maths department

His leadership style was informal, which helped produce an atmosphere in which mathematical research flourished. By the time Warwick accepted its first students in October 1965, the department was already competing with other universities at an international level. The glass building it is now housed in is named after Zeeman in honour of his tremendous effort in founding the department.

Zeeman left Warwick in 1988, and was made an honorary professor there upon his departure. He moved on to become the principal of Hertford College, Oxford and Gresham professor of geometry at Gresham College, London. He retired from these two positions in 1995 and 1994 respectively.

Catastrophe

From 1966 to 1967, Zeeman was a visiting professor at the University of California, Berkeley. Shortly after his return to Warwick, a dynamical systems symposium was held, attended by many of the world leaders in dynamical systems, including Stephen Smale and René Thom. They inspired his change of discipline from topology to dynamical systems, and prompted Zeeman to spend a sabbatical with Thom in Paris, where he grew fascinated with what came to be known as catastrophe theory.

Cardboard catastrophe machine

The bifurcation set on my cardboard catastrophe machine

He is famous for inventing a catastrophe machine, consisting of a circular disc that can rotate freely about its centre, and two elastic bands of identical length attached on the edge of the disc  The other end of one piece of elastic is fixed, while that of the second elastic is free to move on the plane. Zeeman’s machine has some surprising behaviour: as the free end moves around, the disc would do something unexpected: it flips to a drastically different position. The flipping action is a vivid example of a catastrophe: a discontinuous effect resulting from a continuous change of forces. You can plot the set of points at which the disc flips, called the bifurcation set, which takes on a diamond-like shape consisting of four concave edges and four cusps.

According to Hirsch, Zeeman once tried to take his machine with him to the USA. As soon as he mentioned the name of his machine, US customs cleared the room and had Zeeman arrested!

Zeeman played a huge role in making catastrophe theory a hot topic in the 70s. He was keen to apply it in numerous contexts such as nerve impulses, the collapse of bridges, stock markets and even prison riots. On returning to Warwick, he taught a course in catastrophe theory for undergraduates, which soon became extremely popular.

Outreach and the Royal Institution

Zeeman was not only passionate about his research, he was also heavily engaged in promoting mathematics to the general public. He was the first mathematician to present the Royal Institution Christmas lectures, in 1978. Going by the title of Mathematics into pictures and including a mix of pure and applied mathematics, Zeeman inspired his live audience with the aid of various demonstrations, including his own catastrophe machine. His lectures sparked plenty of enthusiasm; among the live audience was Marcus du Sautoy, a budding young mathematician who would go on to deliver his own Christmas lectures in 2006.

But that was not the only result from Zeeman’s Christmas lectures—they also served as the inspiration behind the Royal Institution masterclasses for both mathematics and engineering. Starting from 1981, the masterclasses were designed to inspire keen schoolchildren across the UK. When I attended them as a schoolgirl in 2005, I had no idea about the history behind the masterclasses at the time!

Awards and positions

In 1975, Zeeman was elected a fellow of the Royal Society and was president of the London Mathematical Society (LMS) from 1986 to 1988. He also took up many other positions and received various awards—too many to list in one article.

Zeeman was knighted in 1991 for his “mathematical excellence and service to British mathematics and mathematics education”. More recently, the Institute of Mathematics and its Applications (IMA) and the LMS jointly set up the Zeeman medal in his honour to recognise those who “have excelled in promoting mathematics and engaging with the general public”. And you don’t have to be a seasoned professor to earn the medal—the 2016 Zeeman medal was won by author Rob Eastaway.

Epilogue

Zeeman had three daughters and three sons from two marriages. One of his daughters, Mary Lou Zeeman, became a mathematician herself, eventually collaborating with her father in mathematical ecology.

Part of Zeeman’s (academic) family tree

In total, Zeeman had 29 PhD students, including David Epstein, Terry Wall and Jaroslav Stark, and over 700 other descendants, including me!

It is now about two years since Zeeman passed away aged 91, on 23 February 2016. I never met him in person, but I have seen and felt the effects of his legacy, and am proud to be (academically!) descended from him. Some of his methods in dynamical systems which he applied in mathematical ecology came in handy for my research in population genetics.

Now I want to share my love of mathematics with the rest of the world, like Zeeman did, and there is no better place to start than Chalkdust. If you’re equally inspired, you should write for Chalkdust too! I look forward to reading it.

post

Biography of Sophie Bryant

In 1884, Sophie Bryant’s paper, On the ideal geometrical form of natural cell structure, was published by the London Mathematical Society (LMS). It was ambitious, logical and descriptive: it looked at the phenomenon of the honeycomb.

Her insight was that the complex and beautiful honeycomb shape was a product of the natural activity of bees. All that was needed was for each bee to excavate its own cell at approximately the same rate as the others, and to use the excavated material to build up the walls of its cell. Bryant’s conclusion, that elongated rhombic semi-dodecahedra are the natural form of honeycomb cells, had been observed by Kepler.

A rhombic semi-dodecahedron can be made by putting square-based pyramids on the faces of a cube

In the eighteenth century, it was believed that the honeycomb was the most efficient cell shape possible, but this is now known not to be the case. In 1964, the Hungarian mathematician Fejes Tóth observed in his paper, What the bees know and what they do not know, that there are in fact more efficient cell shapes which have yet to be determined.

Kepler conjectured in 1611 that no packing of balls of the same radius in three dimensions has density greater than the face-centred cubic packing—the cannonball packing—with a density of about 74%. Bryant’s paper assumed this conjecture to be true, as it had appeared obvious for centuries and many had attempted proofs.

Cannonball packing. Image: Wikimedia Commons user Greg L, CC BY-SA 3.0

The conjecture was eventually proved by Hales et al in 1998. Their computer-assisted proof was so huge that it took 12 referees to check
it. After five years, the referees said that they were 99% sure that the proof was correct. Unusually, Annals of Mathematics published the paper in 2005 without complete certification from the referees. It was finally accepted as proven in 2014, and then only with the aid of massive amounts of computer time.

Bryant’s approach to the subject was not unusual at that time. Abstract proofs, so essential to us now, were not as common as general discussion of mathematical phenomena. She wrote “The form of a natural structure is a logical result of its mode of genesis, and that form is ideal of which the mode of genesis is perfectly regular”. She states that there are only three possible arrangements without explaining why these are the only ones.

Bryant’s paper is notable since it is the first published paper written by a woman member of the LMS. However, she was not the first woman to be elected to membership, being preceded by two remarkable women, Charlotte Angas Scott and Christine Ladd Franklin.

Charlotte Angas Scott

Though Bryant was the first woman member of the LMS to publish a paper, she was not the first woman member. That honour goes to Charlotte Angas Scott (1858—1931), an algebraic geometer, who became a member in 1881.

Scott had been aided in her mathematical education by an enlightened father. This resulted in her obtaining a scholarship to Girton College, Cambridge. However, women in Cambridge were not granted degrees until 1948 and she had to be content with the accolades of her peers.

She was appointed a lecturer at Girton and received an external BSc degree from London University, and later a doctorate. Scott moved to the newly opened Bryn Mawr College for women in the USA where she was appointed head of mathematics, and remained for forty years.

Being first

Being the ‘first woman’ was not unusual for Bryant. She was the first woman to receive a DSc degree in England, studying what was then mental and moral philosophy, but today would be referred to as psychology and ethics. She was also one of the first three women to be appointed to a Royal Commission—the Bryce Commission on Secondary Education in 1894–95—and she was one of the first three women to be appointed to the senate of London University.

While on the senate she advocated setting up a day training college for teachers, which eventually became the Institute of Education. Later in 1904, when Trinity College, Dublin opened its degrees to women, Bryant was one of the first to be awarded an honorary doctorate. In Cambridge, she was also instrumental in setting up the Cambridge Training College for Women which eventually became Hughes Hall, the first postgraduate college for women in Cambridge.

She was also, it seems, one of the first women to own a bicycle.

Beginnings and early widowhood

Bryant was born in Ireland, and was fortunate to learn mathematics as well as other academic subjects with her five siblings in a very natural way from their father, the Rev WA Willock DD. A keen educationalist, he had been a fellow and tutor at Trinity College, Dublin and had gained high honours in mathematics and mental sciences.

When Bryant was about thirteen, her family moved to England and her family education continued until she attended Bedford College, where she was awarded the Arnott scholarship for science in 1866. She sat the Cambridge local examination for girls in 1867 and was the only one to be placed in the first class of the senior division.

In 1869, Bryant married the surgeon Dr William Hicks Bryant, only to be widowed the following year when he died of cirrhosis at the early age of 30.

Christine Ladd Franklin

The second woman member of the LMS, who also joined in 1881, was Christine Ladd Franklin (1847—1930), an American mathematical logician.

Though Johns Hopkins University was not open to women, UCL’s JJ Sylvester, then professor of mathematics, urged not only that she be admitted, but arranged for her to do graduate work under his supervision and to be granted a fellowship. As Johns Hopkins did not award degrees to women, she left without a PhD for her dissertation on symbolic logic.

She was finally awarded a PhD by Johns Hopkins forty-four years after she submitted her dissertation, when she was seventy-eight years old.

Schoolteacher and doctorate

After a short interval, Sophie Bryant returned to her studies. While she had been sitting her examinations, she was introduced to Frances Buss, the headmistress and founder of North London Collegiate School (NLCS), an excellent school then and still highly regarded today. It had been founded in 1850, the year of Bryant’s birth.

Bryant arranged to meet Buss who, in 1875, invited her to teach mathematics at NLCS and encouraged her to take a training course as well. Three years later, London University opened its degrees to women. As Bryant had not had a conventional education, she had to learn Latin and biology to matriculate before she could sit for her degree. In 1881, she earned a BSc degree, gaining a first class in mental and moral science and second in mathematics.

In 1884, she received a science doctorate. The NLCS, where she had continued to teach, presented her with scarlet doctoral robes. Bryant was influential in improving the education system and introduced a scheme of enlightened and serious study.

In 1885, Buss died and Bryant became the headmistress of NLCS until her retirement.

Psychology

Bryant receiving her
doctorate. Image: North London Collegiate School

Meanwhile, she continued to publish ambitious papers. In her 1884 paper in Mind, The double effect of mental stimuli; a contrast of types, Bryant attempted to analyse the difference between reflex actions, which are performed without conscious thought, and consciously controlled actions.

She was grappling with a contemporary problem: the understanding of consciousness. Unfortunately, her arguments are too diffuse to shed much light on the problem.

In 1885, she published a paper in the Journal of the Anthropological Institute, Experiments in testing the characters of school children. This study, undertaken at the suggestion of Francis Galton, produced an early account of the use of open-ended psychometric tests to deduce personality types. Bryant claimed that her results agreed with the observations of teachers familiar with the children but did not provide any supporting evidence. Despite incomplete analysis, this was a pioneering study.

Later life

Bryant was interested in Irish politics, and wrote books on Irish history and ancient Irish law. She was an ardent Protestant Irish nationalist and was active in the Home Rule movement, which pressed for Irish self-government within the United Kingdom. She wrote on women’s suffrage in 1879 but later advocated postponement until women were better educated in politics.

She enjoyed mountain climbing and she summited the Matterhorn twice. Her death in 1922 was both tragic and unexpected. Only four years after retirement, she was on a mountain hike near Chamonix, in France, when she went missing. Her body was found thirteen days later with several injuries.

Epilogue

Although Bryant’s direct contribution to mathematical scholarship was not substantial, her influence as a teacher and educationalist was immense. The rising number of women mathematicians today is a lasting tribute to her work.

post

Roots: Blaise Pascal

The influence of Blaise Pascal is most keenly felt in his work on probability and the binomial theorem, illustrated by the famous Pascal’s triangle. It cannot be denied that Pascal’s triangle is a thing of mathematical beauty. However, this array of numbers was not discovered by its namesake, rather its applications and importance were highlighted by Pascal in his work, akin to Pythagoras’ theorem, which was certainly not invented by Pythagoras himself. However, Pascal’s legacy to mathematics goes further than the instantly recognisable triangle …   Continue reading

post

Florence Nightingale, statistician

In her copy of Thomas à Kempis’ fifteenth century The Imitation of Christ is the inscription, in her own writing, “I only wish to be forgotten”. Although remaining a very private figure throughout her lifetime, this was not a desire that could ever be fulfilled. Since her death on 13 August 1910, at the age of 90, Florence Nightingale’s fame and the legend of the lady with the lamp has failed to dim. That her name has become a synonym for nursing is, however, misleading; and the public perception of her as a nurse does a great disservice to her life as a pioneering statistician.
Continue reading

post

Hedy Lamarr: Hollywood star and secret inventor

The golden age of Hollywood was a time of classic movies and classic movie stars. A time of ‘frankly my dear’, ‘play it again’ and ‘whoops Mr Parson’ (I made the last one up). Yet only one star was billed as ‘The most beautiful woman in the world’. This was Hedy Lamarr: an Austrian-born actress, former wife of an arms dealer, international movie star, and occasional inventor. Her most celebrated invention was something without which today’s mobile phone and Wi-Fi technology would not be possible: frequency-hopping.  Continue reading

post

The Mathematical Games of Martin Gardner

It all began in December 1956, when an article about hexaflexagons was published in Scientific American. A hexaflexagon is a hexagonal paper toy which can be folded and then opened out to reveal hidden faces. If you have never made a hexaflexagon, then you should stop reading and make one right now. Once you’ve done so, you will understand why the article led to a craze in New York; you will probably even create your own mini-craze because you will just need to show it to everyone you know.

The author of the article was, of course, Martin Gardner.

Continue reading

post

Klaus Roth

Chalkdust is very sad to hear that the 1958 Fields medallist Klaus Friedrich Roth, who was featured in our first issue, passed away on the night of the 9th/10th November in Inverness, Scotland. Born in what was then Prussia in 1925, he spent most of his life in the United Kingdom, graduating with a BA from Peterhouse College, Cambridge, in 1945 and obtaining an MSc (1948) and PhD (1950) from University College London. In 1958, whilst at UCL (1946–66), he was awarded the Fields medal for solving “in 1955 the famous Thue-Siegel problem concerning the approximation to algebraic numbers by rational numbers and [proving] in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turán of 1935)”. In 1966, he was awarded a chair at Imperial College London, where he remained for the rest of his career, retiring in 1988 (although he remained there as a visiting professor until 1996).

You can read more about Klaus Roth and his work on the Thue-Siegel problem here.

post

John Forbes Nash: the legacy

When describing John Forbes Nash, Jr (13 June 1928 – 23 May 2015), it’s hard to be more succinct than Richard Duffin, a professor at the Carnegie Institute of Technology, who wrote, in his letter of recommendation to Princeton, that ‘this man is a genius’. It was 1948: Nash, having abandoned a degree in Chemical Engineering for one in Mathematics, was only just embarking on a journey that would ultimately make him one of the most famous mathematicians of the 20th Century. Despite the interest of Harvard University, Nash eventually decided to pursue his graduate studies at Princeton and it was there that he published the 317 word paper, Equilibrium points in N-person games, that introduced the Nash Equilibrium and won him the Nobel Prize for Economics (jointly with Reinhard Selten and John Harsanyi) in 1994. As a result of this work in game theory, Nash was appointed to the RAND Corporation, which applied this relatively young field to the pressing policy issues of the time: nuclear weapons, the space race, the Cold War.

Continue reading