The name Gladys West is probably unfamiliar, but she was part of creating something you probably use often enough: GPS. You wouldn’t think a child who grew up on a sharecropping farm would wind up as an influential mathematician, but perhaps watching her father work very hard for very little and her mother working for a tobacco company made her realize that she wanted more for herself. Early on, she decided that education was the way out. She made it all the way to the Naval Surface Warfare Center.
While she was there she changed the world with — no kidding — mathematics. While she didn’t single-handedly invent satellite navigation, her work was critical to the systems we take for granted today.
If you were to travel back in time to the turn of the previous century and try to convince the average person that the grains of sand on just about any beach would be the basis of an industry worth hundreds of billions of dollars within 100 years, they’d probably have thought you were crazy. Aside from being coarse, rough, and irritating, sand is everywhere, and convincing anyone of its value would be a hard sell, unless your interlocutor was a real estate visionary with an appreciation of the future value of seaside property and a lot of patience.
Fast forward to our time, and we all know the value of the material that comes from common quartz sand: silicon, specifically the ultra-purified crystals of silicon that end up as the wafers we depend on to build the circuitry of life. The trip from beach to chip foundry is a long and non-obvious one which would not have been possible without the insights of an undistinguished Polish student and one-time druggist who discovered the process that made the Information Age possible: Jan Czochralski.
In my youth I worked for a paid ambulance service, and while we all lived for the emergency calls, the routine transports were the calls that paid the bills. Compared with the glamor and excitement of a lights-and-siren run to a car wreck or heart attack, transports were dull as dirt. And dullest of all were the daily runs from nursing homes to the dialysis center, where rows of comfy chairs sat, each before a refrigerator-sized machine designed to filter the blood of a patient in renal failure, giving them another few days of life.
Sadly, most of those patients were doomed; many were in need of a kidney transplant for which there was no suitable donor, while some were simply not candidates for transplantation. Dialysis was literally all that stood between them and a slow, painful death, and I could see that at least some of them were cheered by the sight of the waiting dialysis machine. The principles of how the kidneys work have been known since at least the 1800s, but it would take until 1945 for the efforts of a Dutch doctor, using used car parts and sausage casings, to make the predecessor of those machines: the first artificial kidney.
Whether you are young, old, or a time traveling Vulcan, something unites all of us globally: the innocent LEGO blocks that encourage creativity over spoon-fed entertainment. Have you noticed the excess of zombified children and adults alike drooling over their collective screens lately? Back in the ancient times, all a child needed to create hours of joy were plastic interlocking bricks and a place for their parents to trip over them. The LEGO Group harbored the inspiration of our childhood inventiveness, and none of it would have been possible without the founder, Ole Kirk Kristiansen (or Christiansen). The humble carpenter from Denmark forever made his mark on the little Scandinavian country, one brick at a time.
Well, maybe not at first. You see the plastic LEGO bricks we all know and love were initially made of wood. And they were also not actually bricks.
It may be hard for those raised on cinematic video games to conceive of the wonder of watching a plain white dot tracing across a black screen, reflecting off walls and bounced by a little paddle that responded instantly to the twist of a wrist. But there was a time when Pong was all we had, and it was fascinating. People lined up for hours for the privilege of exchanging a quarter for a few minutes of monochrome distraction. In an arcade stuffed with noisy pinball machines with garish artwork and flashing lights, Pong seemed like a calm oasis, and you could almost feel your brain doing the geometry to figure out where to place the paddle so as not to miss the shot.
As primitive as it now seems, Pong was at the forefront of the video game revolution, and that little game spawned an industry that raked in $108 billion last year alone. It also spawned one of the early success stories of the industry, Atari, a company founded in 1972. Just last week, Ted Dabney, one of the co-founders of Atari, died at the age of 81. It’s sad that we’re getting to the point where we’re losing some of the pioneers of the industry, but it’s the way of things. All we can do is reflect on Dabney’s life and legacy, and examine the improbable path that led him to be one of the fathers of the video game industry.
As computer networks get bigger, it becomes increasingly hard to keep track of the flow of data over this network. How do you route data, making sure that the data is spread to all parts of the network? You use an algorithm called the spanning tree protocol — just one of the contributions to computer science of a remarkable engineer, Dr. Radia Perlman. But before she created this fundamental Internet protocol, she also worked on LOGO, the first programming language for children, creating a dialect for toddlers.
When a 13-year old Marie-Sophie Germain was stuck in the house because of the chaotic revolution on the streets of Paris in 1789, she found a refuge for her active mind: her father’s mathematics books. These inspired her to embark on pioneering a new branch of mathematics that focussed on modeling the real world: applied mathematics.
Post-revolutionary France was not an easy place for a woman to study mathematics, though. She taught herself higher maths from her father’s books, eventually persuading her parents to support her unusual career choice and getting her a tutor. After she had learned all she could, she looked at studying at the new École Polytechnique. Founded after the revolution as a military and engineering school to focus on practical science, this school did not admit women.
Anyone could ask for copies of the lecture notes, however, and students submitted their observations in writing. Germain got the notes and submitted her coursework under the pseudonym Monsieur Antoine-August Le Blanc. One of the lecturers that she impressed was Joseph Louis Lagrange, the mathematician famous for defining the mathematics of orbital motion that explained why the moon kept the same face to the earth. Lagrange arranged to meet this promising student and was surprised when Germain turned out to be a woman.
Gauss and Germain
‘Le Blanc’ also corresponded with German mathematician Carl Friedrich Gauss on number theory. When Napoleon’s armies occupied the town the famous mathematician lived in, Germain enlisted a family friend in the army to check that Gauss had not been harmed. Gauss didn’t realize who had helped him out, until he discovered that ‘Le Blanc’ was Sophie Germain, he wrote to her thanking her for her concern and praising her mathematical prowess given the hurdles set before her.
“How sweet is the acquisition of a friendship so flattering and precious to my heart. The lively interest you took during this terrible war deserves the most sincere recognition….But when a person of this sex, who, for our mores and prejudices, must recognize infinitely more obstacles and difficulties than men to become acquainted with these thorny searches, knows how to get rid of these obstacles and to penetrate what they have, most hidden, must undoubtedly, she has the most noble courage, talents quite extraordinary, genius superior.”
As well as working on the thorny and theoretical problems of number theory, Germain worked on applying mathematics to real world problems. One of these was a challenge set by the Paris Academy of Science to mathematically describe the elasticity of metal plates. An experimenter called Ernest Chladni had demonstrated that a metal plate would resonate in odd ways when vibrated at certain frequencies. If you put sand on the plate, it would collect in different patterns created by the resonance of the plate, called Chladni figures. To win the prize, the solution had to predict these figures.
The Mathematics of Stress and Strain
Mathematically predicting the behavior of metal plates could make it easier to design metal objects and predict how they would behave under stress. The prize was set in 1808 but was so difficult that Germain was the only one who decided to try to solve it, as it required coming up with a whole new way to analyze and describe how materials bend and change under stress.
The first two solutions that she submitted were rejected due to mathematical errors, but the third version won her the prize in 1816. However, due to the Academy policy of not allowing women to join (and to only attend events if they were wives of members), Germain was not able to attend the ceremony where the prize was granted. She was also not allowed to attend meetings of the Academy. After the Academy failed to publish her prize-winning work, Germain had to pay to publish the work herself in 1821.
Later, her friend Joseph Fourier allowed her to attend meetings and presentations, but the mathematical establishment never really accepted her, or her work. In a letter to a colleague in 1826 she complained about the way they rebuffed her:
“These facts are my domain and it is to me alone that they remain hidden. That’s the privilege of the ladies: they get compliments and no real benefits.”
In the same letter, Germain complained of suffering fatigue and she was diagnosed with breast cancer shortly afterwards. She died in 1831. Her final years were spent working on a solution to Fermat’s Last Theorem, and just before her death she published a partial solution that was the basis for much research into the theorem, which was finally solved only with computer help in the late 1990s.
Although Sophie Germain never earned a degree in her lifetime, she was given an honorary degree in 1837 from the University of Göttingen at the suggestion of Gauss, who noted that
“she proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree.”
The Academy that snubbed her also now offers an annual prize for mathematics in her name. Perhaps more importantly, her work formed the basis of the study of elasticity and stress in metals that allowed engineers to build larger objects and buildings. Creations such as the Eiffel tower in 1887 were directly influenced by her work, and it laid some of the groundwork for Einstein’s theory of General Relativity.
Modern scholars argue that Germain could have been more than she was: her work, they argue, was hamstrung by a limited understanding of some of the fundamental concepts that Gauss and others had described. Although her work was fundamental and important, if she had been given free access to the education that she wanted and deserved, it’s easy to imagine that she would have gone farther.
The first two solutions that she submitted were rejected due to mathematical errors, but the third version won her the prize in 1816. However, due to the Academy policy of not allowing women to join (and to only attend events if they were wives of members), Germain was not able to attend the ceremony where the prize was granted. She was also not allowed to attend meetings of the Academy. After the Academy failed to publish her prize-winning work, 
