Joseph Fourier and the Sinusoids

Joseph Fourier, linocut, 11" x 14", by Ele Willoughby, 2024

Sometimes people ask to commission a scientist portrait of someone I had already considered portraying. I don't think you can study physics for as long as I did and not think about Joseph Fourier. So when asked to make his portrait, I had a plan in mind.

French mathematician and physicist Jean-Baptiste Joseph Fourier (1768-1830) is best remembered for his work Fourier series, Fourier transforms, Fourier’s law of conduction, Fourier analysis and harmonic analysis and their use to solve heat transfer problems. He is also often credited with proposing the greenhouse effect as early as 1824.

Born in Auxerre, a son of a tailor, he was orphaned at age nine. Recommended to the Bishop of Auxerre, he was educated by the Benedictine Order of the Convent of St. Mark. Being a commoner he could not seek a commission in the scientific corps of the army but took a military lectureship in mathematics. He took a prominent part in promoting the French Revolution in his district. Nonetheless he was briefly imprisoned by the Terror. In 1795 he was appointed to the École Normale and later succeeded Lagrange at the École Polytechnique. After accompanying Napoleon as a scientific adviser on his Egyptian expedition in 1798 he was appointed secretary of the Institut d’Egypte, where he organized workshops for the French to make munitions, wrote several papers for the Egyptian Institute (now Cairo Institute). He returned to France in 1801 after the British victories, to resume work at the École Polytechnique but Napoleon appointed him Prefect of the Department of Isère in Grenoble. In Grenoble he began to experiment on the propagation of heat, and contributed to the comprehensive catalogue Description de l’Égype.

In 1820 he published his theorem on polynomial real roots, that a polynomial with real coefficients has a real root between any two consecutive zeros of its derivative. (This is technically a corollary of a theorem published by Budan in 1807 and 1811, also now known as Fourier’s theorem). In 1822 he succeeded Delambre as Permanent Secretary of the French Academy of Sciences, and he published his heat flow research in his Théorie analytique de la chaleur (The Analytical Theory of Heat) which included his important claim that any function of a variable, continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. The idea is true (with some conditions, later discovered by Dirichlet) and a breakthrough. This is the foundation for what we call the Fourier transform, an invaluable tool in math, physics and engineering. During the 1820s he realized that the Earth, at its distance from the Sun should be colder is it is only heated by solar radiation. He wrongly thought Earth has must then receive significant radiation from interstellar space but he did consider the correct answer that the atmosphere acts as an insulator. He referred to experiments by Saussure who measured temperature in a vase with several inset panes of glass, who noted that the inner chambers (under more glass) become hotter. Fourier noted that if the atmosphere makes a layer like the glass in the vase we would get the same effect on Earth (though he noted that Saussure’s experiment did not account for the convection we get in the atmosphere). This is now recognized as the first conception of what we know as the greenhouse effect.

He became a foreign member of the Royal Swedish Academy of Science in 1830. He experienced heart aneurysms in both Egypt and Grenoble and died 16 May 1830. He was buried in Père Lachaise Cemetary in Paris in a tomb with an Egyptian motif and his name is one of 72 inscribed on the Eiffel Tower.

My portrait (inspired by the portrait by Boilly) alludes to Fourier decomposition: the sine waves sum to the violet waveform framing him. With an infinite series of sines, his very outline, as a function of x (the horizontal axis across the page) could be made. Likewise the circles in the background are another visualization of building up an image of Fourier himself from sine waves. If you trace the perimeter of a circle on a continuously advancing sheet of paper you produce a sine wave. (Mathematicians and physicists often use this equivalency to think of a real sinusoid as the real part of a complex function where a vector formed by by its real and imaginary parts traces a circle around the origin). So a simplified function tracing the main lines in Bouilly’s portrait can be produced by the sum of sine waves produced by tracing the series of these epicycle circles. 

There are a lot of great tutorials about this online, but a particular shout out to 3Blue1Brown on YouTube who not only has an excellent series of videos (it's everything you ever wanted to know about Fourier series but were afraid to ask) but he's also specifically made animations of producing a path that approximates Bouilly's portrait with rotating vectors or Fourier epicycles here.

Kitchen Table Mathematics and the Pentagonal Tilings of Marjorie Rice

 

Marjorie Rice and her Pentagonal Tilings, linocut, 11" x 14", by Ele Willoughby, 2023

This is a linocut print of Marjorie Rice (née Jeuck, 1923–2017) who discovered four new pentagonal tilings of the Euclidian plane. Each print is 11" x 14" and printed by hand. Each of her 4 new pentagonal tilings is shown and the design on her shirt is based on her own "Butterflies" tessellation using the Type 13 pentagon (with two right angles and two sides of the same length). I made this portrait for the "pattern" prompt of #printerSolstice.

Born 100 years ago, she grew up poor in Florida, attending a one-room school house, where she skipped two grades, but college was never an option for her. She married conscientious objector Gilbert Rice in 1945 and they moved to Washington, D.C., where he worked in a military hospital and she worked as a commercial artist. They moved to San Diego with their infant son, who did not survive infancy, but went on to have five more healthy children.

Rice had no training as a mathematician, having completed half of a correspondence art course after high school, but she was always interested in mathematics, puzzles and art. She began to follow Scientific American writer and amateur mathematician Martin Gardner's monthly column, rushing to devour the magazine before her son (who had the subscription) could get it. Gardner had reported in July, 1975, in his article, "On Tessellating the Plane with Convex Polygon Tiles," that mathematician Richard Brandon Kershner had finally completed the task of finding all the remaining convex polygons which could make tessellations to tile the plane, an open mathematical problem which had intrigued thinkers since Ancient Greece. But by the next month, one of his readers, Richard James III, had reported that he had found a new convex pentagon tiler. Rice was inspired to start her own search. 

She spent her spare time discretely drawing diagrams on the kitchen table when no one was around which she hid when her husband, kids or friends stopped by. Her daughter assumed she was simply doodling, not discovering new mathematics. She succeeded in her search by February, 1976, and wrote Gardner of her new pentagon type and its variations in shape with which she could tile the plane. Lacking any math education, she had also developed her own notation system to describe shapes. Gardner forwarded Rice's letter to mathematician and tiling pattern expert Doris Schattschneider. Though she was skeptical of Marjorie's odd and unique notation system, which she likened to "hieroglyphics" Schattschneider was able to validate Rice's results. Rice did not stop there. By October she had found 58 pentagon tilings (most previously unknown) of two paired pentagons which could tile "transitively" which she categorized in 12 classes. By December she had found two more new convex pentagon tilers and 75 distinct tessellations by pentagons that were in blocks that could be seen as "double hexagons". By the following December (1977), she found her fourth convex pentagonal tiler and 103 "2-block transitive" pentagon tilings. Through the next decade she continued to find more pentagonal tiling patterns and explored aperiodic tilings and she used her discoveries to make Escher-like tessellation patterns of flowers, shells, butterflies and bees overlain on the geometric shapes (which can still be seen on her website).

Martin Gardner included Rice's discoveries in a collection of his columns 'The Mathematical Gardner', in 1988 and Doris Schattschneider’s article 'In Praise of Amateurs.' Though she was too shy to ever publish her own results or give any talks about her work, Schattschneider convinced Rice and her husband to attend her talk about Rice's work at  a meeting of the Mathematical Association of America held in Los Angeles, in 1995. When she was pointed out in the audience, she received a standing ovation! In 1996, she was the subject of a documentary on CBC's 'The Nature of Things.'

The Mathematical Association of America in Washington, D.C. had one of the ceramic tiles of one of the tilings discovered  installed in the foyer of the headquarters of 1999. Rice's papers and research notes are preserved at the Eugène Strens Recreational Mathematics Collection at the University of Calgary Library. Michaël Rao, with a computer-aided mathematical proof in 2017, established that there are a total of precisely 15 families of convex pentagons, including Rice's 4. He employed her same approach: considering how corners could come together at vertices. Rice was tickled to be recognized and loved being able to discover something new, long sought after by mathematicians 

Sources

Marjorie Rice, Wikipedia, accessed February, 2023

Marjorie Rice, Intriguing Tessellations, accessed February, 2023

Natalie Wolchover, Marjorie Rice's Secret Pentagons, Quanta Magazine Abstractions Blog, July 11, 2017

Anna Weltman, Marjorie Rice, Inspired by Math, and Subways, Math Munch, February 25, 2013

Beatrice "Trixie" Worsley - One of the World's First Computer Science Doctorates

Trixie Worsley, linocut 11" x 14" by Ele Willoughby, 2018

Trixie Worsley earned one of the very first doctorate in computer science, supervised by Douglas Hartree and Alan Turing at Cambridge. Amongst the first computer scientists in Canada, she was certainly the first woman in the field here. She focused on writing software, development of computer libraries, scientific computation and was co-author of the first compiler Transcode (vital to physicists) as well as teaching in the new field of computer science. Her work provides insight into the history of the nascent field of computer science. She published her computational insights and solutions for problems in physics, biology and computer science.

Beatrice Helen Worsley (1921-1972), a quiet and accommodating girl known as Trixie, was born in Queretaro, Mexico, to English parents who had moved to Mexico so her father could work in her mother's family's textile mill. This mill had been destroyed by rebels in 1917, and Trixie's future parents had had to move again so her father could instead work for Rio Grande group’s CIMSA mills. Trixie's mother home-schooled her and her older brother, and the family remained cut off from the local community for safety during this turbulent time in Mexico's history. In 1929, the family moved to Toronto, mainly for the sake of the children's education. Trixie attended Brown public school for a few years before going to the private Bishop Strachan School, where as the top student of her day, she excelled in the university track classes. She graduated with honours in 1939, winning awards for math and science and the Governor General's Award for the highest grades in the school. She entered Trinity College at the University of Toronto with both a general proficiency entrance award and the Burnside Scholarship in Science. Getting top marks in most of her classes she won more scholarships and transferred to Math and Physics in 1940, specializing in applied math. She first saw computers during a summer job in the actuarial department of the Manufacturers Life Insurance Company in Toronto in 1942.

As soon as she completed her bachelor's in mathematics at the University of Toronto in 1944, she enlisted in the Women’s Royal Canadian Naval Service, known as the Wrens. As a researcher at the Naval Research Establishment at Her Majesty’s Canadian Ship Establishment Stradacona, in Halifax, NS, she focused on harbour defence. Worsley was one of 6 Wrens amongst 50 scientists, officers and support staff, who were responsible for things like degaussing ships to limit their magnetic signature and vulnerability to German magnetic mines, torpedo guidance and researching different techniques for harbour defence. Most of this group left the service within a year of the end of the war to pursue further education, with the special opportunities offered to veterans. A small number including Worsley, the only remaining Wren, stayed on. The newly promoted Lieutenant Worsley began researching the badly understood electrochemistry of hull corrosion in 1945, performing experiments at sea. She set the Wrens' record for time at sea, at 150 days, including during the rough mid-winter months. Her endurance and knowledge earned her the respect of the crews, even doing what she called a man's job. She demobilized in 1946 to pursue further education.

She went to MIT for a master's in math and physics with Henry Wallman, at the Radiation Laboratory, where she was exposed to computers and wrote a thesis called Mathematical Survey of Computing Devices with an Appendix on Error Analysis of Differential Analyzers. There were only really a handful of computers out there at the time, in universities, industry and national labs and she surveyed the literature on all of them as well as those planned, and began to be an expert on this new technology and to wish for a future in the field.

Returning to Canada in 1947, where there were not yet any available jobs in computers, she worked at the National Research Council (NRC) in Ottawa as an aerodynamics research officer in the mechanical engineering division, but it didn't hold her interest. She managed to negotiate a move to the new University of Toronto Computation Centre - the only existing Canadian computer R&D program - as one of two project assistants in 1948 (hired for $200 month). She and her colleagues worked on tabulating results for Atomic Energy Canada (AEC) in Chalk River. Worsley used Meccano to build her own differential analyzer with small improvements on the design published by Douglas Hartree and Arthur Porter in 1935. This was a cheap device, one of about 15 ever built worldwide, that was accurate enough to solve many scientific problems. Worsley and the other project assistant were sent to Cambridge to learn about EDSAC at the Cambridge Mathematical Laboratory in 1949. Worsley co-wrote the first program to run on EDSAC and her report on the machine's first run was published in the proceedings of a meeting on high-speed computing at Cambridge that June. She didn't return to Canada because she started her doctorate at Newnham College, with Douglas Hartree as supervisor (with Alan Turing and Maurice Wilkes). She returned to Canada before completing her dissertation but math professor Byron A. Griffith agreed to supervisor her till she completed her work. Hartree approved her thesis, Serial Programming for Real and Idealized Digital Calculating Machines, and she was awarded a PhD in 1952 and then published her first scientific paper. Hers is amongst the first doctorates awarded involving modern computers.

The FERUT computer from the Univerisyt of Toronto Archives

Meanwhile, the ambitious (and expensive) design and construction project of Canada’s first computer, the University of Toronto Electronic Computer Mark I (UTEC) was underway, aiming to produce a nationally-shared university, government and military resource. Plagued by mechanical problems and tube failure, the head of the AEC argued that funding should be withdrawn and moved to purchasing a completed system, the Ferranti Mark I in 1952. Worsley herself named the machine FERUT for “Ferranti computer at the University of Toronto,” and operated the new machine, creating software for everything from problems in atomic physics to the St. Lawrence Seaway calculations. With her colleague Gottlieb she taught courses on its use, but it was notoriously difficult and almost all their students quit. Worsley then worked with physics professor J.N. Patterson Hume, to write a compiler called Transcode, to make higher-order programming possible - a much easier method, if slower to execute. Allowing users to program in a language, rather than machine code, and enter numbers in decimal, rather than binary, hugely simplifier their task and had a huge impact on computing in Canada. It allowed dozens of research groups nationwide to use FERUT to solve a wide-array of scientific problems. Transcode was an immediate success and hundreds of people learned it before the FERUT was replaced with an IBM 650 in 1958. Despite her education, publications and teaching track-record, Worsley was repeatedly passed over for promotion and received less recognition than her (male) peers. She was only promoted from Computation Centre mathematician to assistant professor of physics in 1960. Worsley published more papers than any of the other staff of the Computation Centre in the 1960s as her career veered towards teaching. She was promoted to associate professor of physics and computer science when U of T started a graduate department of computer science in 1965.

She left U of T in 1965 to join Queen's University Computing Centre in Kingston, Ontario, likely influenced by her slow career advancement and treatment as a woman researcher at U of T. It caught her colleagues by surprise as she was moving to a University without a computer science program, only an outdated IBM 1620 and would be computing advisor to the Computing Centre with teaching duties, but not a professor. But, she was lured there to start the new Computing Centre and start anew. A new department of Computing and Information Science was created in 1968 complete with master's program thanks largely to Worsley's efforts and she was promoted to associate professor. In 1971 she took a sabbatical at the Department of Applied Analysis and Computer Science at the University of Waterloo to study assembler coding and computer architecture, but she had a fatal heart attack at age 50 on May 8, 1972.

After her untimely death, Worsley left her entire estate to Cambridge University to set up the Lundgren Fund, in honour of Helge Lundgren, for doctoral math or science students with preference to those in computer science "whose research has been interrupted by national service or personal misfortune." Scott M. Campbell who wrote the great biography of Worsley for the IEEE has been unable to identify this Helge Lundgren, though he suspects this might be tied to whatever it was that drew Worsley suddenly back to Canada before she had finished her doctorate.

Her report on the first run of the EDSAC was included in Brian Randell's classic 1973 book, The Origins of Digital Computers, leading to posthumous fame in the history of computer science. The Canadian Association of Computer Science / Association d'Informatique Canadienne (CACS/AIC) honoured her with a lifetime achievement award in 2015, alongside her former colleagues Hume and Gottlieb. A second Canadian woman supervised by Douglas Hartree at Cambridge, professor emerita of Vanderbilt University Charlotte Froese Fischer established a U of T computer science graduate scholarship in Worsley’s name, for doctoral candidates who have taken an active role in promoting women in the field of computer science. Froese Fischer met Worsley when they both worked in the Computing Centre and remembers her insights, wry humour and "way of expressing herself in a few memorable words.”

I was surprised how much of Trixie's career I could relate to directly. I've spent much more than 150 days doing research at sea, but never (in the Northern Hemisphere) later than a fairly miserable, snowy October. I have a pretty good sense of what it would have been like on the North Atlantic in the mid-winter. Her work as a WREN was on electrochemistry of hull corrosion. As someone who has done marine electromagnetics, I know this is still an active research topic for my naval colleagues. Her doctoral thesis included a great variety of problems, but one, the calculation of second-order corrections to the value of gravity from pendulum measurements at sea is something I know about as my own doctorate involved adapting a gravimeter to use at sea. I've shown Trixie based on a photo of her seated in front of the FERUT, and other images of the machine. I was influenced by my former colleague at U of T Physics, Gordon West's descriptions of the first computers they used and peering directly into the memory of the machine themselves through its output oscilloscopes. MIT makes her master's thesis available, so I looked it up but ended up opting to try and allude to her career as a programmer (rather than a hardware researcher) by reproducing the structure of a flow chart from one of her papers about computation methods in atomic physics.

References
Nina Haikara, 'Honouring Canada's first female computer scientist: U of T's Trixie Worsley,' U of T news, May 26, 2015

Scott M. Campbell, "Beatrice Helen Worsley: Canada's Female Computer Pioneer," IEEE Annals of the History of Computing, volume 25, no. 4 (Oct-Dec 2003), p.51--62.

Beatrice H. Worsley, 'The Self-Consistent Field with Exchange for Neon by FERUT Program,' Can. J. Physics, Vol. 36, 1958

Beatrice Worsley, The Canadian Encyclopedia, accessed April 20, 1018

Smillie, Keith (2002). "Beatrice (Trixie) Worsley". The Computer and Me - A Restrospective Look at Some Computers and Languages.

Beatrice H. (Trixie) Worsley | CS-CAN | INFO-CAN, accessed April 20, 2018

Inspired by Science on The Etsy Blog

The Etsy blog just posted Karen Brown's article featuring 5 Etsy artists who are inspired by science, including me!

Lise Meitner and Nuclear Fission Linocut History of Physics by minouette

 

“I think the idea that art and science are separate is unfounded,” says print maker Ele Willoughby of minouette. “It takes creativity to be a good scientist and experimentation to be a good artist.” In her Etsy shop, Ele explores art and science through a series of portraits of scientists inspired by the bi-monthly challenges of the Mad Scientists of Etsy team. “I love hearing from parents who want to inspire young children with portraits of scientific heroes or heroines,” she says.

There are some fabulous artists in that inspiring intersection of art and science, and several of my prints included.

Hypatia (Part 1)

Hypatia was the amongst the earliest woman mathematician recorded in Western history. This is my imagined portrait of her in a first edition linocut print in copper and purple-blue on lovely Japanese kozo (or mulberry) paper, 12.5 inches by 12 inches (31.7 cm by 30.5 cm). There are 12 prints in the edition.

Hypatia lived in the 3rd century AD in Alexandria, Egypt, which was part of the Roman Empire. She was born at some time between about 350 and 370 and died in 415 C.E. She was the head of school, where she taught Neoplatonist philosophy, astronomy and mathematics. She believed in empiricism and natural law. She was the daughter of a famous mathematician, Theon Alexandricus (ca. 335–405), with whom she worked and published edited versions of Classical texts in mathematics. She also pursued her education in Athens and Italy before returning to Alexandria.

It is difficult to precisely specify her own contributions to mathematics, since many documents have been lost and it was the tradition of the time to write commentaries on the great texts of Classical mathematics. These were essentially edited or annotated versions, which may have included corrections and new material, but which were known by the name of the original author. Also, her work and her father's work may be confused and conflated. This confusion of attribution is common for female scholars of the Classical world. It is known that she wrote commentaries on 13-volume Arithmetica by Diophantus, the Conics of Apollonius, and edited Ptolemy's Almagest and on Euclid's Elements. I've alluded to these in my print; the diagrams on her table, from right to left, show conic sections, Hipparchus' construction to calculate the distances to the Sun and Moon (which was part of Ptolemy's Almagest), and some basic Euclidean geometry. She is also credited with a more efficient means of doing long division - which is no mean feat when you recall she was using Roman numerals!

She charted heavenly bodies. She built and instructed her pupils in the design and use of the astrolabe. The astrolabe, like the one she is holding in the print, was central to astronomy and navigation from antiquity through the Renaissance. It allowed people to measure the inclination of celestial bodies and could be used to locate or predict the positions of the Sun, Moon, planets, and stars, determine local time given local latitude and vice-versa, or for surveying and triangulation. You can see the outer circle is notated with the 24 hours of the day (in Roman numerals) and the inner circle with the symbols for the contellations of the Zodiac. Thus, she was not only a mathematician, but what we would today call an astronomer, a physicist and an engineer.

She met a terrible end. She became embroiled in a dispute between the Prefect and the Christian Patriach, later known as Saint Cyril, who spread rumors that she was a witch who had cast spells on the Prefect. Scolasticus, a contemporary Christian writer, wrote that she was kidnapped on her walk home, taken to a church, where she was stripped and torn limb from limb and then burned by an angry mob of Christians lead by a fanatic. News of her murder lead the community to be condemned. Scholasticus comments, "Surely nothing can be farther from the spirit of Christianity than the allowance of massacres, fights, and transactions of that sort".

Because of her violent and scandalous death, her story has been used by many groups of people with many competing interpretations of its meaning. She has been claimed by competing sects of Christian (who either defend or defame her) throughout history, Pagans, Wiccans, and feminists as a symbol. Somewhat ironically, though she was accused of being a Pagan "devoted at all times to magic, astrolabes and instruments of music, and she beguiled many people through her Satanic wiles" by the 7th-century Egyptian Coptic bishop John of Nikiû, she was later employed by Christians as a symbol of Virtue. She has appeared in literature, art, graphic novels (Alan Moore's Promethea) and cinema (including Agora (2009), by Alejandro Amenábar, staring Rachel Weisz) . Her murder has been called the first witch hunt, the end of Classical Antiquity and the beginning of the Dark Ages - but these claims are not well-founded in the actual historical events.

Chances are that many of these proponents of Hypatia and claims by various groups that she was one of their own are not really representative of the historical Hypatia, so much as her symbolic power. There is no unbiased contemporary evidence that she was either a true Pagan or a witch. She was a Neoplatonist, which was both a spiritual and philosophical tradition, so not truly a staunch secularist either. For all the competing Christian claims (Catholic versus Orthodox, Protestant versus Catholic, churches of Constantinople versus Alexandria) or the view that she was a tragic victim of the Church's conquest of science of the time, it seems likely that she was a political pawn in a battle between Patriarch and Prefect. But, it is very hard not to see that from the earliest known instance, a woman who excelled in science and math was the object of jealously, treachery, accusations of witchcraft and, in far too many cases, violence and murder. Though, she was not the only Classical woman in philosophy and mathematics and those who came before and those who followed did not share her terrible fate.

It was her fate which lead to her lasting fame but it is her achievements that should be remembered. One contemporary claimed she outshone her father, regarded the best mathematician of his generation. Perhaps most importantly, she preserved the mathematical heritage of ancient Greece and Roman and was talent and respected teacher. She was known to wear the robes of a scholar, usually reserved for men, as was the role she took as philosopher, mathematician and advisor to the leaders of her day. As Scholasticus wrote:

"There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner which she had acquired in consequence of the cultivation of her mind, she not infrequently appeared in public in the presence of the magistrates. Neither did she feel abashed in going to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more"

[Edited for accuracy February 3, 2022]

Alan Turing and the Enigma

This is a linocut portrait of Alan Turing (1912-1954). It is printed in indigo and orange by hand on lovely Japanese kozo (or mulberry paper), 9.25" by 12.5" (23.5 cm by 32 cm) in an edition of eight.

This year we celebrate the centenary anniversary of Alan Mathison Turing, OBE, FRS (23 June 1912 – 7 June 1954), British mathematician, cryptanalyst, computer scientist, prophet and hero. He is someone I've long thought to portray, but I have been stumped. My scientists are shown with images of something quintessential to their science, or the reason they are famous (or should be), but Turing had so many accomplishments, it wasn't obvious what to portray or how. You might recall his portrayal in Neal Stephenson's 'Cryptonomicon'. I was introduced to him many year ago by Douglas Hofstadter's 'Gödel, Escher, Bach'¹. You may also be familiar with the Turing Test or at least its portrayal in Blade Runner. Turing foresaw not only that machines might quite likely develop the capacity to think (after all, our brains are only made of matter, and complex systems of neurons, which either fire or not, much like an electronic switch), but that we needed an objective, double-blind test to determine whether something/someone was able to think, as early as 1950, when most people were only dimly aware of the existence of any sort of computer. But Turing quite literally defined what we now mean by computation itself (with his concept of Turing Machines) back in 1936. During the WWII he worked as a codebreaker and invented the device which was finally able to crack the notorious German cryptographic Enigma machine (in its more complex later incarnation)! His work undoubtedly saved many lives, and today we recognize him as a genius and a hero. In my print, I've included a simplified diagram of the mechanism behind the Enigma with its rotors or scramblers which acted as monoalphabetic substitution ciphers, literally scrambling letters at each turn. During, his all too short life, he also made important contributions to mathematical biology and explaining morphogenesis (the biological process that causes an organism to develop its shape) and the existence of Fibonacci numbers in biology. To indicate this later work, I've made the pattern of his tie look like the sort of Turing pattern produced by reaction–diffusion systems. This work presaged much later work in chaos theory.

Tragically, he lived in a time even more biased and bigoted than our own. Rather than recognizing the magnitude of his contributions to society during his lifetime, he was prosecuted for his homosexuality (still illegal in Britain in 1952) and forced to undergo chemical castration. He died two years later, after eating a cyanide-poisoned apple (determined by the coroner to be a suicide). It is truly abominable they way he was treated; while we can't address the past injustice we can remember, recognize and celebrate his remarkable achievements today.

There are many serious looking photos of Turing. I chose one of him smiling as inspiration for this portrait. He clearly enjoyed his work, and had a sense of humour (evident in the silly names he gave mathematical techniques he discovered)², so I chose to remember him laughing.  

¹I cannot recommend this book strongly enough. Read it. Immediately, if not sooner.
² i.e. Banburismus and Turingery

Happy Pi Day!

Who doesn't love pi, or π (or pie for that matter)? On March 14 (which can be written 3/14*) we celebrate the world's most famous irrational number, so central to geometry, trigonometry, calculus, Fourier Analysis... you name it. If it's mathematical, it needs more π.

Dinosaur Comics

The video was produced by Austin-based musician Michael John Blake via the New Scientist magazine.

*Though I prefer the proper, metric date version, 14/3/12.