Philippa Garrett Fawcett, Mathematical Tripos, ad eundum, isotropic helicoid, Édouard Lucas, Mersenne primes, Lucas Tower

Hi, I'm Ana, the Ethical Systems Nerd, with the April 4th episode of Nerd Nodes.

Today is the birthday in 1868 of Philippa Garrett Fawcett, who is best known for doing really well on an exam. The exam in question was the Mathematical Tripos, or the final exam in a Cambridge undergraduate mathematics degree. Fawcett, as a woman in 1890, was permitted to attend Cambridge and to sit the exam, but she wasn't allowed to be granted a degree, or to be named the Senior Wrangler, the traditional title for the man who achieved the highest exam score in a given year. So, when Fawcett's score on the Tripos turned out to be 13% higher than the second highest score, she was proclaimed to be "above the senior wrangler", to the great excitement of the assembled crowd and to a chorus of newspaper headlines the following morning. The University of Cambridge would never grant her a degree, but since her exam grade was at least officially recorded, she was able to claim an ad eundum degree from sister school Trinity College, Dublin after that university started admitting women as full members in 1904, making her part of a group of over 700 women known as "steamboat ladies" for the means of transportation used to claim their degrees.

The Universities of Cambridge and Oxford are each made up of many constituent colleges and Fawcett had been an undergraduate at Newnham College, a college for women which her mother helped to co-found in 1871, whose students were granted the right to sit University exams in 1881, and which achieved full status as a college of the university in 1948. I attended Cambridge for a year and my college, Caius, first admitted women in 1979, the year I was born. Fawcett was hired as a mathematics lecturer by Newnham after she... didn't... graduate, and stayed there for 10 years, doing mathematical research in the field of fluid dynamics.

One of her published papers deals with a theoretical shape called an isotropic helicoid, proposed by William Thompson or Lord Kelvin in 1871. William Thompson, incidentally, was Second Wrangler in his year at Cambridge. An isotropic helicoid is any shape which experiences an equal amount of fluid drag no matter what its orientation is. Thompson hypothesized that an isotropic helicoid should spin when placed in liquid. Recent experiments have used 3-D printers to print shapes which should be isotropic helicoids, but if they do spin then it's too small to observe, at least so far. You can download plans to print your own isotropic helicoid at the Thingiverse, a website filled with 3d printable shapes, many of them creative commons licensed.

Another mathematician born on April 4th was the French mathematician Édouard Lucas, born in 1842. If you have ever committed yourself to a project perhaps without really realizing how long it would take, then you'll relate to Lucas who set out at the age of 15 to prove that the 39 digit Mersenne number 2^127 - 1 (170141183 4604692317 3168730371 5884105727) was prime. He would spend 19 years verifying by hand that 2^127-1 was in fact a Mersenne prime, proving it in 1876. This remained the largest known prime for 75 years, and will probably always remain the largest prime number proven by hand.

The next two prime numbers were discovered on the same day, Jan 30 1952, with the aid of the Standards Western Automatic Computer. So-called because it was built by the U.S. Bureau of Standards, and located on the West Coast of North America in Los Angeles, California. A mathematician named Raphael M. Robinson would use an improved version of Lucas' technique, along with the computing power of the SWAC, to find the next 5 Mersenne primes over the year 1952.

In addition to number theory, Édouard Lucas also enjoyed recreational mathematics. You have probably seen some version of the Tower of Hanoi game, which was invented by Lucas and turned into a physical packaged game for sale in 1883, with the tagline "A GAME BROUGHT BACK FROM TONKIN", Tonkin being the French name for northern Vietnam, which the French Third Republic was in the process of invading and occupying in 1883, an invasion which no doubt helped France forget the humiliation of the Franco-Prussian War and the Siege of Paris in 1870. Lucas himself had been an artillery officer in the French army. So, Lucas capitalized on the orientalist Zeitgeist and named his game the Tower of Hanoi, giving it a fake backstory, attributed to a fake professor at a fake university, named using anagrams of his own name and academic affiliation. Lucas's boxed version of the game contained eight disks, which need to be moved between three pegs, following the rule that a larger disk can never be placed on top of a smaller disk. It turns out that the minimum number of moves to win is 2^n − 1, where n is the number of disks. So at least 255 moves are required to solve a Tower of Hanoi game with 8 disks. Lucas's fake backstory alludes to a myth that the game, with 64 golden jewel-encrusted disks, has been played at a Brahmin temple for many years, and when the game is finally complete, this will bring about the end of the world. It's easy to imagine how Lucas, who spent 19 years manually verifying the primality of the 127th Mersenne number, would have related to this story. He offers a prize of "a million francs and more" to anyone who completes the 64 disk version of the game by hand, before announcing that it would require 18 446 744 073 709 551 615 moves, or approximately 5 billion centuries, so rest assured that if the game does bring about the end of the world, it will be a long time from now. The game has remained in use as a recreation as well as a psychological assessment tool, and its mathematical underpinnings have applications in a wide variety of fields. It also appears in a variety of science fiction stories, and in the 2011 movie Rise of the Planet of the Apes, it is aptly renamed as a Lucas Tower.

 

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