Historically, mathematics has been sadly lacking in diversity of role models, and while it’s much better now, it’s still not great (the list of HLF laureates this year is depressingly homogeneous). People who have made huge contributions to mathematical endeavour over the centuries – female mathematicians, mathematicians of colour and those in oppressed minorities – have often been denied proper credit for the work they have done.

Last month, mathematics celebrated the 100th birthday of one of its most brilliant and successful proponents – Katherine Johnson, whose achievement of reaching the age of 100 pales in comparison to the work she has done in her lifetime, and the barriers to success she’s faced and overcome during that time.

Katherine Johnson

Born in West Virginia on 26th August 1918, Katherine was a promising mathematician from a very early age, completing high school at 14 and graduating with degrees in mathematics and French at 18. This was during the time of racial segregation in the USA, so her schooling took place in segregated ‘black-only’ schools and colleges, but she received expert tuition from brilliant professors, including chemist and mathematician Angie Turner King, and W. W. Schiefflin Claytor, the third ever African American to receive a PhD in mathematics. Johnson was a brilliant student, completing all the math courses offered by the college, to the point that Clayton created extra courses – including one on analytic geometry – just for her.

After working as a teacher, raising three children and caring for her terminally ill first husband, she wanted to go into research mathematics, and in 1953 was offered a job by the NACA (National Advisory Committee for Aeronautics), later to become NASA. Katherine worked as a computer – literally, as part of a team which manually computed and verified the answers to mathematical problems, before digital computers were good enough to do this reliably. The team processed data from black box recorders, and analysed topics such as gust alleviation for aircraft – studying how to make corrections to the flight controls to account for incoming uneven winds, and maintain equilibrium in unstable conditions.

Katherine’s brightness was recognised by her superiors, and she was reassigned to the Guidance and Control Division of Langley’s Flight Research team, and then later to the Spacecraft Controls Branch, working with a team of otherwise exclusively white, male engineers. Segregation laws were still in place at this time, meaning Johnson had to use separate dining areas and toilets than her colleagues.

She also felt discrimination due to her gender – despite working on papers equally with others, it took a fight to get her name added as an author, and hers was the first woman’s name to be included as co-author on a published technical report in the Space Flight Division. By being assertive and standing their ground, as well as delivering brilliant work, Johnson and her colleagues managed to change the status quo and get more recognition and respect.

During her career, Johnson worked on many crucial space missions, including the 1961 flight of Alan Shepard – the first American in space, as well as on Apollo 11 (the moon landing) and Apollo 13. Katherine calculated the trajectory for Shepard’s first space flight, and the launch window for his Mercury mission later that year. Astronaut John Glenn was so impressed with her work, he requested she personally verify the calculations on his orbital mission. The calculations she was doing, some of which incorporated early use of digital computers, allowed teams to quickly find capsules after they landed, based on accurate trajectory calculations.

Euler’s Method

In the 2016 film Hidden Figures, which tells the story of Johnson and her colleagues at NASA, as well as the struggles they faced, one gripping scene shows Johnson arguing for the use of Euler’s method to calculate the flight trajectory. One of the engineers describes this method as ‘ancient’ – it was published by Euler in one of his books in the late 1700s, so that’s a fair assessment – but Johnson defends the method as suitable for what they need.

Differential equations describe complex systems, like a capsule moving through space – there will be many gravitational fields acting on it at once, as well as the forces acting on it due to its own motion. The coupled differential equations describe the system, and how all the forces interact, in terms of a set of variables.

The solution to a differential equation will be a curve – a path described by an equation relating all the relevant variables – and it’s often not possible to get an explicit function that describes this curve by looking at the differential equations themselves. For simple examples, this can be done using calculus, but in general it can be impossible.

Euler’s method exploits the fact that on small scales, it’s possible to calculate numerically what the curve looks like in very small sections, using the differential equations. By taking a set of points along the length of the curve, and at each point getting the position of the curve and the angle of its slope, it’s possible to reconstruct the curve – by connecting the dots – with a reasonable degree of accuracy. This was Johnson’s suggestion: using centuries-old mathematics to put humans in space.

The scene features prominently in the film, despite containing some pretty serious mathematics, which is unusual for a Hollywood blockbuster. Rudy Horne, one of the mathematics advisors on the film, described it to the director who was so inspired by this display of Johnson’s brilliance, he immediately included it in the script.

Katherine Johnson has received many awards and honorary doctorates in recognition of her achievements, and in 2017 the Katherine G Johnson Computational Research Facility in Hampton, Virginia was named in her honour. She was awarded the Presidential Medal of Freedom in 2015, as a pioneering example of an African-American woman in STEM. West Virginia University have an endowed STEM scholarship in her name, and a life-sized statue of her on campus, which was unveiled on her 100th birthday.

Johnson is a shining example of mathematical brilliance, and we are lucky that her mathematical skills were accompanied by dogged persistence and a willingness to push back against the oppressive system she found herself in. There must have been many more examples of amazingly talented mathematicians who didn’t get the same opportunity to use their abilities in this way, and if we can endeavour to offer opportunities to everyone equally, hopefully we’ll see many more people, and projects like the space programme, succeed.