The time of the Renaissance resonates with the names of individuals who have made an impact in the development of mathematics. One name, however, echoes more loudly than the others: Gerolamo Cardano. Continue reading
This year, on 26 August, one of the most memorable and well-known mathematicians, Katherine Coleman Goble Johnson, celebrated her 100th birthday. This is a tribute in honour of her life so far.
Family life and first steps towards mathematics
Katherine was born on 26 August 1918 in White Sulphur Springs, a small town in West Virginia. She was the youngest of four children and was always the smart kid-she finished high school at the age of 14 and earned her Bachelor of Science in mathematics and French from the West Virginia State University at the age of 18. This, in part, was thanks to her father, who moved the family closer to a school to help his children get a better education. She still remembers her family dearly. Especially fondly, Katherine talks about her father’s stepmother, known as granny. They would visit her house to eat some of her delicious pancakes-just as anyone would with their grandmothers! The four children loved their parents very much: they thought of their mother as the prettiest lady in the world and their dad as the most handsome man. Katherine says she was daddy’s girl, but she always remembers her father telling her, “you are as good as anybody in this town, but you’re no better”.
Katherine was always good at mathematics. She has memories from childhood very clearly linking her to the subject: “I counted everything. I counted the steps to the road, the steps up to the church, the number of dishes and silverware I washed… anything that could be counted, I did”. But mathematics wasn’t the only subject for her. Sure, she loved it, but she was just as good in English because it also felt logical to her.
When asked why mathematics became the subject she was most fond of, she says it was because it was the subject you had to work hard for. It was the one subject with a right and a wrong, and once you got it right, it was right-unlike history!
The next push towards maths came at university. In eighth grade she had a maths teacher who happened to teach at the university Katherine went to years later. One day, she happened to meet the teacher again, who told her “if you aren’t in my math class this semester, I’m coming after you!” So Katherine had no choice but to go to maths class and her career in mathematics had begun. Later at university, her maths interests were taken care of by Mr Claytor. He added courses to the university almost exclusively for Katherine because he could see her potential. It was he who steered her towards research mathematics and eventually NASA.
Female mathematician at NASA
Katherine started her career in a rather unusual manner-she became a computer. Back then, these were the people who did the calculations for NASA’s predecessor NACA (the National Advisory Committee for Aeronautics) before the space race began. Katherine was sent to the flight research division.
She counts her character as one of the key things that contributed towards her career as a NASA mathematician. She remembers how her siblings and parents always used to try to shush her because she was always so assertive and stubborn. After she was invited to join the men doing the mathematics behind the calculations she would perform, she started fighting for her own place within the team. She demanded to see all the data, and asked to join the confidential meetings NACA held, slowly gaining respect in the midst of the all-male team.
Especially noted in the recent film Hidden Figures are the times when Johnson overcame and battled racism, as the only female and the only African American in the department. Katherine herself always says, though, it was never anything special: she just did her job and was appreciated for that, not her sex or skin colour.
On 5 May 1961, Freedom 7 was to be launched. Modern technological computers were already running the numbers and everything was being prepared for the mission that would make Alan Shepard the first American in space. The astronaut, however, was feeling uneasy. He made the call to NASA to speak to Katherine Johnson. Would she redo the calculations?, he asked. Because if she got it right, he knew it was right-he would feel safe to go on the mission. Katherine approved the computer’s calculations. She admits the NASA team was much more worried about Shepard never making it back to Earth. If he missed the trajectory by a few degrees or tried entering at a different velocity, he would never return home.
The mission was successful and Katherine later also checked the trajectory for the Apollo 11 and Apollo 13 missions, further proving her incredible mathematical skills.
As a token from NASA, Katherine received an American flag that flew to the moon.
At this point you may wonder-wait, you’re telling me about all these amazing things she has done, but you aren’t sharing any of the maths. Unfortunately, many of her articles aren’t available to the public. However, the three that are, are described below. I must warn you-all contain sophisticated mathematics and will most certainly take quite a while to wrap your head around.
However, I can give you some insight into what the three papers contain. The first paper (Skopinski & Johnson, 1960) focuses on calculating the azimuth angle when placing a satellite over a predetermined position to ensure safe landing.
The second paper (Westrick & Johnson, 1962) is an analysis of the data from the Echo 1 satellite. It contains a lot of very nice graphs-it’s worth having a look just for the curves!
The third paper (White & Johnson, 1964) probably has the toughest maths, but similar to the first one, it focuses on finding solutions of some variables for the landing of a satellite. Arm yourself with some patience-the papers are worth your time even if they might seem a bit daunting at first!
She retired from NASA in 1986, but she still has her hands full. For 50 years she enjoyed singing in a church choir. She loves playing bridge and other mathematical games, and she plays the piano and enjoys spending time with her six grandchildren and 11 great-grandchildren. She has authored or co-authored 26 research papers, and she has worked on the space shuttle and Project Apollo’s lunar lander. For her achievements and lifelong work, she received the Presidential Medal of Freedom in 2015. In 2016, the BBC named Katherine in their 100 Women 2016 as one of the most inspiring women alive. And then there’s the aforementioned film Hidden Figures, which revealed the story behind the three brilliant mathematicians Dorothy Vaughan, Mary Jackson, and of course Katherine, which has since conquered my and many other mathematicians’ hearts.
Katherine Johnson has been an inspiration for mathematicians all around the world, showing how one person can change so much. From gaining respect in one of the most prestigious research facilities in the world in times of unimaginable discrimination, to creating mathematics which helped many astronauts find their way back home; from simply being a wonderful person to being an incredibly talented mathematician-here’s to Katherine Johnson on her 100th birthday!
- Hidden Figures, directed by Theodor Melfi.
- TH Skopinski, Katherine G Johnson, Determination of Azimuth Angle at Burnout for Placing a Satellite Over a Selected Earth Position, September 1960.
- Gertrude C Westrick, Katherine G Johnson, Orbital Behavior of the Echo I Satellite and its Rocket Casing During the First 500 Days, June 1962.
- Jack A White, Katherine G Johnson, Approximate Solutions for Flight-Path Angle of a Reentry Vehicle in the Upper Atmosphere, July 1964.
When I first came across the great Black mathematician and statistician, David Blackwell (1919-2010), circa 1975, I was actually informed that he was white. He was also then Irish. Or, so I was told by a triumphal fellow MSc economics and econometrics student at Southampton University, himself Irish, and now also a professor of economics.
The occasion of this initial meeting with Blackwell was our econometrics class’s introduction to the eponymous Rao-Blackwell theorem—a fundamental result in the theory of optimal statistical estimators. In simple terms, this theorem shows how to improve upon a rudimentary unbiased estimator of a statistical parameter, and indeed, get the best unbiased estimator of that parameter, when certain technical conditions are satisfied. I remember being struck by the beauty of this result. Perhaps it was my excitement about it that led my Irish colleague to try to deflate me by claiming his own racial and national part-ownership for the theorem by telling me that Blackwell was a white Irishman—Rao’s Indian extraction being self-evident. Maybe, more charitably, he was just engaging in supposedly characteristic Irish blarney, without malice. Regardless, I never bothered to check his claim—and, why should I have doubted a fellow student’s word about something as inconsequential as someone’s nationality, as I thought then?
So, for almost a decade afterwards, I happily persisted in the belief that Blackwell was indeed Irish and blithely assured others of this. I must have given much wry amusement to those who knew otherwise. It was not until the academic year 1984-85, which I spent as a joint fellow at CORE (Centre for Operations Research and Econometrics) and IRES (Institut de Recherches Économiques et Sociales) at Université Catholique de Louvain-la-Neuve, that I was finally disabused of my misinformation by another researcher. Continue reading
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.
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.
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.
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.
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.
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.
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.
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.
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.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.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.
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.
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.
PsychologyMeanwhile, 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.
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.
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.
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
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.
In 2017, a new polymer £10 note will enter circulation in Scotland, featuring the portrait of a Scottish mathematician and scientist: the honour chosen by public vote.
At a time when democracy and equality are being fervently debated, this issue’s Roots column focuses on the legacy of Mary Somerville.
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