Polymath Claude Shannon and Theseus the Electromechanical Mouse in a Maze

 

Claude Shannon, linocut print, 11" x 14" by Ele Willoughby, 2026

I was recently commissioned to make a portrait of Claude Shannon. It seemed clear that it would need some binary to represent his work, and the math and computer science in particular. I opted to also include Theseus in his maze, which hits on the electrical engineering, and the humour in what he did.

This is my linocut portrait of Claude Elwood Shannon (1916-2001), mathematician, electrical engineer, computer scientist and cryptographer credited with laying the foundations for the Information Age. Each hand-printed linocut on 11" x 14" Japanese kozo (or mulberry paper) shows Shannon in front of binary numbers and with his electromechanical mouse Theseus with its maze. Though partially behind him, the binary numbers represent the standard ASCII code for "CLAUDESHANNON". 

At the University of Michigan, he took two degrees: one in electrical engineering and the second in mathematics, graduating in 1936. His MIT masters thesis is arguably one of the most impactful ever completed. In it, "A Symbolic Analysis of Relay and Switching Circuits", he showed that we can construct any logical numerical relationship through the electrical applications of Boolean algebra. This is now the underlying theoretical basis for   digital computing and digital circuits. He completed his doctorate at MIT in 1940 on mathematical genetics.

During WWII he worked in the field of cryptanalysis for US national defence, doing fundamental work on codebreaking and secure telecommunications, and writing a paper which is considered one of the foundational pieces of modern symmetric-key cryptography.

His 1948 paper "A Mathematical Theory of Communication" laid the foundations for the field of information theory. This paper was important to the invention of the compact disc, the internet, mobile phones and even our understanding of blackholes. He introduced the term "bit," invented the signal-flow graph and co-invented pulse-code modulation and the first wearable computer. 

In 1950, he designed and built a learning machine, with the help of his wife mathematician and research collaborator Betty Shannon. They built an adjustable maze with sensors (an electromechanical relay circuit) which followed the path of an electromechanical mouse whimsically named Theseus. Theseus could search corridors until it found a target (the penny in my portrait). Then, the mouse could be moved to anywhere in the maze. If the location was known, it would go immediately to the target. If the location was unknown, the mouse would search until it found a known position, adding this knowledge to its memory, and proceed from the known position. This way it could eventually find the most direct route. This is the first known artificial learning device.

In 1951 he joined the CIA's  Special Cryptologic Advisory Group. He was a professor at MIT from 1956 to 1978. His work was also foundational for artificial intelligence. He was a co-organizer of the 1956 Dartmouth workshop, considered to be the discipline's founding event. He published papers on the programming of chess computers. His Theseus machine was the first electrical device to learn by trial and error, being one of the first examples of artificial intelligence.

Wang Zhenyi astronomer

Wang Zhenyi, linocut, 11" x14" by Ele Willoughby, 2026

For the 12th and final #PrinterSolstice2526 prompt multiplication, I have chose Qing era polymath - astronomer, mathematician, meteorologist and poet - Wang Zhenyi (王贞仪), who amongst her accomplishments in her brief life, was a book on mathematics for young readers with simplified rules for multiplication and division, to help children learn even those she lived when formal education for girls was rare. In the absence of any known portraits, I made my print after researching Qing Dynasty pavilions, lanterns, tables and mirrors, along with women's fashions and available astronomical instruments in China in the 1790s.

Astronomer, mathematician, meteorologist and poet Wang Zhenyi (1768-1797) left her impact on Qing era China during her short life. Raised by her father and grandparents, her ancestral home was in Anhui province. Her grandfather, Wang Zhefu, an avid reader and collector of books, was governor of Fengcheng County and the Xuanhua District. When her father, Wang Xichen failed the imperial examination to enter the civil service, he decided instead to study medicine, and wrote a 4-volume Collection of Medical Prescriptions. While many Qing women were denied literacy and education, Zhenyi absorbed it all: astronomy from her grandfather, poetry from her grandmother Dong, medicine, geography and mathematics from her father. When she was 9, her grandfather died and the family travelled to Jilin near the Great Wall to mourn and attend his ornate funeral. They remained there for five years. While there, Zhenyi and three other upper-class girls studied under the Lady of Bu Qianyao, and she was able to read her grandfather's library with its full 75 bookcases. She also learned to ride, do archery on horseback and martial arts from the wife of the Mongolian General Aa. She became an expert mounted markswoman. On her own, she began further exploring mathematics and astronomy, reading Chinese texts, and Western classics like Euclid's Elements.

Zhenyi then travelled with her grandmother and father, visiting Beijing, Shaanxi, Hubei, Guangdong and Anhui. This unusual experience exposed her to more history and more breadth of society than was common for most young women. She and her father travelled south of the Yangtze river, before moving back to the capital and settling in Nanjing when she was 16. At 18, her poetry united her with female scholars in Jangling, and she began to focus on mathematics and astronomy. At 25, deemed late to marry at the time, she married Zhan Mei from Xuancheng in her home province. Their marriage was happy but they had no offspring. She gained fame from her poetry, mathematics and astronomy knowledge. She advanced meteorology to work on weather forecasts for farmers. She even took on some male students, an extraordinarily uncommon thing for a young woman scholar at the time. 

Flemish Jesuit missionary and polymath Ferdinand Verbiest (1623-1688), known as Nan Huairen in Chinese, became Director of the Imperial Astronomical Bureau, introduced updated armillary spheres, sextants and celestial globes to China and had them adorned with Chinese motifs. He reformed the inaccurate Chinese calendar and diplomatically fostered Sino-European scientific exchange.

 
The Qing Dynasty closed door policy meant that astronomy in China was isolated from advancements that were being made in the West. Jesuit missionaries had shared the works of Copernicus, Galileo and Brahe. Copernicus' model was shared as a useful tool rather than a doctrine, so they could both avoid Church ire and respect local traditions. However, Newton's Principia, including his universal law of gravitation, was not available in a Chinese edition until 1850. When Wang Zhenyi took pains to explain how we could live on a globe without falling off, she was doing so without benefit of an explanation of gravity.

She wrote Dispute of the Procession of the Equinoxes using her observations of celestial phenomena, explaining how equinoxes, the two days each year in the spring and fall when days are nights are equal length, move and how to calculate their movements. Using both her observations and astronomical texts, she wrote about the number of stars, the direction of the revolution of the sun, moon, Venus, Jupiter, Mars, Mercury and Saturn, as well as describing lunar and solar eclipses. Only a small fraction of her works survive. Later articles included Dispute of Longitude and Stars, Explanation of Lunar Eclipses, and Explanation of the Starry Sky. Zhenyi was interested in the causes of lunar eclipses and published her correct understanding in Explanation of Lunar Eclipses. She summarized astronomical theories, from Yu Xi (307-345) who discovered the precession of the equinoxes, to Gui Shoujing (1231-1316) who is credited with inventing the gnomon and a water powered armillary sphere, and managed to calculate the length of a year as 365.2425 days (mere seconds short of the modern value). She connected these scholars' work with Islamic, Western and modern calendars. At a time when many of her contemporaries would attribute an eclipse to the anger of the gods, she bluntly wrote, "In fact, it's definitely because of the moon." She made a demonstration in an outdoor pavilion to show that how these phenomena could be simply explained by the relative motion of Earth (represented by a round table), moon (represented by a mirror) and sun (represented by a suspended round crystal lamp). Moving these three, as celestial bodies would move she could explain lunar eclipses. She could show how the Earth's shadow could pass over the Moon. She explained that a lunar eclipse can only happen during a full moon, and a solar eclipse can only happen during a new moon, but only when the alignment is right. Her work also cleared up misunderstandings about celestial mechanics and addressed the gradual shift in stellar positions. She affirmed the Earth is a sphere, writing the Theory of the Earth's Roundness, refuting ideas of a flat Earth. In The Geocentric Theory of the Annual Cycle, she made the case for a heliocentric system. She was a proponent of the Western sun-based, heliocentric calendar over the lunar calendar, for its precision and advocated for its adoption. She argued people needed to be open to new scientific and mathematical ideas, regardless of their origin. 

To improve weather-forecasting she worked on calculating atmospheric humidity. She investigated making better predictions of floods and droughts, understanding that Chinese farmers suffered in extreme weather conditions. 

She mastered the book Principles of Calculation by famed early Qing dynasty mathematician Mei Wending (1633-1721). Knowing the challenge of trying to teach oneself mathematics, she rewrote it in simpler, more accessible language as The Musts of Calculation. To further make mathematics easier for beginners, she developed simplified means of performing multiplication and division. By the time she was 24, this work culminated in her writing The Simple Principles of Calculation. Another math text which she wrote and survives today, is the Explanation of the Pythagorean Theorem and Trigonometry.

She complained that anyone with access to medical books could believe themselves experts and dispense risky medical advice. Though widely read, and taught by her father, she refrained from treating her own ailments. On the other hand, she advised being wary of quacks, and speaking up on if necessary. When her female cousin was given a prescription, she checked her pulse and consulted books on female health. She condemned the physician's prescription as unsuitable for a woman. She emphasized the use of preventative medicine and clearly had practical medical knowledge.

She wrote 13 volumes of poetry which were well-received, praised for their strength and clarity. Her style was not feminine and flowery, as was more common amongst female poets. She wrote about the classics, the history she learned travelling with her father, ordinary working people, the plight of women and the contrast between rich and poor. Her poems showed her compassion for people she encountered. She wrote about the wealthy hoarding rice until it rotted, while the poor faced starvation and how the increasing tax burden impacted rural regions. She faced criticism when she published her poems for pursuing fame and literary writing. She wrote that she "dare not defend herself" since she should adhere to Confucian ethics and its prescriptions for the roles of women, but she also wrote that the classics were intended for both men and women. Thus, it was stubborn and careless to insist that women should not read. She wrote, 

    It's made to believe,

    Women are the same as Men;

    Are you not convinced?

    Daughters can also be heroic?

She pointed out that both women and men, "are all people, who have the same reasons for studying."

She died when she was only 29, likely after a relapse of malaria. Before she died, she entrusted her manuscripts to her friend Qian Yuling, who in 1803 passed them on to her nephew Qian Yiji. He compiled her mathematics texts and wrote a preface praising her achievements. Most of her works have since been lost and are known only through references from other writers. Her work helped bridge the gap between Western and Chinese astronomy and modernize Chinese astronomical understanding. Two hundred years later, her impact is being acknowledged. In 1994 the International Astronomical Union's Working Group for Planetary System Nomenclature approved a small crater on Venus has being named in her honour. We can only wonder what such a prolific and insightful scholar as she might have achieved had she lived a longer life.

References

Astronomy Week 2025: Honouring the Life and Legacy of Wang Zhenyi, School of Mathematics, University of Edinburgh. Accessed March, 2026.

Bernardi, Gabriela. Wang Zhenyi (1768-1797). In: The Forgotten Sisters. Springer Praxis Books(). Springer, Cham. https://doi.org/10.1007/978-3-319-26127-0_23

DeBakcszy, Dale. Champion of Chinese Heliocentrism: The Stellar Mathematics of Wang Zhenyi. The Women in Science Archive. April 25, 2023.

Lutz, R.C. Wang Zhenyi, EBSCO Knowledge Advantage. Accessed March, 2026.

Mehta, Devang. The prolific life of Wang Zhenyi, autodidact, astronomer and poet. Massive Science. November 3, 2017.

Wang Zhenyi (astronomer), Wikipedia, accessed March, 2026

Wang Zhenyi (王贞仪) – Mirror, Wellesley University blog, accessed March, 2026.

Wing-Chung Ho, Clara. The Cultivation of Female Talent: Views on Women's Education in China During the Early and High Qing Periods. Journal of the Economic and Social History of the Orient. Vol. 38, No. 2, Women's History (1995), pp. 191-223 (33 pages)

Yang, Binbin. Guardians of Family Health in Qing China: From the Exemplary Wife to the Reformer. Modern China. Vol. 41, No. 5 (September 2015), pp. 506-538 (33 pages). Published By: SAGE Publications, Inc.

Also, hat's off to this tumblr which carefully presents a full history of Qing Dynasty women's fashions and hairstyles:

https://audreydoeskaren.tumblr.com/post/697018790909820928/abridged-history-of-qing-dynasty-han-womens

Alicia Boole Stott Envisions the Fourth Dimension

Alicia Boole Stott, linocut print, 9.25" x 12.5" by Ele Willoughby, 2026

 The 9th prompt for #PrinterSolstice2526 is volume, so I thought I would portray a mathematician. Not satisfied with regular old three dimensional volumes, I made a portrait of Anglo-Irish mathematician Alicia Boole Stott (1860-1940), so let's talk about four and more dimensions.

Alicia Boole Stott (1860-1940), the third of five daughters, came to mathematics honestly. Her father George Boole, like her, was an autodidact mathematician, philosopher and logician, who served as the first professor of mathematics at Queen's College, Cork, in Ireland and developed Boolean logic, which would later prove so essential to computer programming. Likewise, her mother Mary Everest Boole was an autodidact mathematician and educator, who was further tutored by George and who edited his book on algebraic language, Laws of Thought. When she was only four years old, her father died, and facing poverty, Mary returned to England with Alicia's sisters, where she found a job as a librarian at Queen's College, London and worked as a mathematics tutor. Mary developed her own novel ideas about fostering imagination in teaching mathematics and science to children. She believed that physical manipulation of objects like sticks and stones, or stitching curves onto cards could help children form an understanding of mathematical concepts. She integrated fables, history, philosophy and literature and a fanciful writing style into teaching mathematics to appeal to children. Her ideas on education were not the only way Mary was unconventional; she organized discussion groups with Queen's College students with John Hinton a promulgator of polygamy, much to the disapproval of the authorities.  She was a believer in parapsychology and the occult and was the first woman to join the Society of Psychical Research. She wrote about topics which were controversial at the time, like the occult, eastern philosophy, evolution and animal rights, as well as the pedagogy of mathematics, so most of her books were published long after they were written. 

Unlike her sisters, Alicia stayed in Cork with her grandmother and great-uncle, until she was 11, when she rejoined her mother and sisters in London. Alicia had felt repressed and unhappy living in Cork, but the lodgings her mother could afford in London were "poor, dark, dirty and uncomfortable." The five girls had to share a room. Her mother sold George's Royal Society Gold Medal, to buy a harmonium so they could have music at home. Alicia went to the school attached to Queen's College London. Mary brought up her daughters doing things like projecting shapes of things such as hanging pendulums onto paper "to acquaint them with the flow of geometry." Alicia learned mathematics from her mother and the first two books of Euclid. She did not attend university. Mary had to leave her job as librarian and become a secretary for John Hinton. When she was 17, Alicia returned briefly to Cork, where she worked in a children's hospital but soon returned to London. She learned about higher-dimensional spaces from her future brother-in-law, mathematician Charles Howard Hinton (son of John Hinton). Charles had mystical beliefs about the fourth dimension and thought we humans inhabit a 4D space we will eventually perceive. He had crafted 4D models with hundreds of small coloured cubes of wood, each labelled with its own Latin name, which he shared with the Boole sisters. He used his cubes to try and get the sisters to visualize a 4D hypercube, for which he coined the word tesseract. His cubes became a popular, but notoriously difficult approach to grappling with 4D geometry; Alicia was the only one who mastered them, becoming more adept than Hinton himself. Among other books, Hinton went on to write The Fourth Dimension. 

There are analogues to 3D shapes in 4D. Imagine the five Platonic solids: the cube, the tetrahedron (a pyramid with a triangular base so it has 4 equilateral triangle sides), the octahedron (like two pyramids glued together at the square base so it has 8 equilateral triangles as faces), the dodecahedron (with 12 pentagonal faces) and the icosahedron (with 20 equilateral triangle faces). These are 3D shapes bound with regular polygons with the same number of edges at each vertex. If you relax the rule that the faces must all be the same, you get the collection of semiregular polyhedra. Each of the Platonic solids, along with  the  the familiar convex regular polygons (equilateral triangle, square, regular pentagon, regular hexagon and so on) has what is called a 4-polytope, which is an analogue in 4D where faces are replaced with 3D cells of identical Platonic solids. Similarly, there are semiregular polytopes you can define from the semiregular polyhedra. 

Alicia Boole's model next to a diagram in Pieter Schoute's diagram
of the same 3D cross-section of a 4D polytope

Alicia discovered that there are exactly 6 regular convex 4-polytopes. Swiss mathematician Ludwig Schläli had beat her to this discovery, in 1850 before she was born, but he had not published his work. He had submitted his manuscript for publication, but it was rejected as it was too lengthy. It was finally published 6 years after he died. He named these shapes polyscheme, but Alicia introduced her term polytope (based on the German term polytop) as she was unaware of Schläli's work (and her term was the one adopted). American mathematician Washington Irving Stringham rediscovered 6 regular convex 4-polytopes and published his results in 1880, but Alicia was unaware of his paper and working in isolation. In 1884, English theologian and schoolmaster Edwin Abbott Abbott published a satirical novella called Flatland: A Romance of Many Dimensions in which lines and 2D polyhedral characters living on a plane encounter a 3D sphere; as it moves through the plane the sphere appears as cross-sections, a sequence of growing and shrinking circles. Analogously, Alicia was able to visualize 4-polytopes as 3D cross-sections, allowing her to unfold a 4D problem into 3D problems. Simply using Euclidean constructions and synthetic methods (the only tools at her disposal, having never learned analytic geometry) she was able to produce 3D central cross-sections of all 6 regular polytypes in 4D, and make coloured drawings and cardboard models of each. Of these 6, 5 are 4D analogues of the Platonic solids (the hypercube or 8-cell, the hyperoctahedron or 16-cell, the hypertetrahedron or 5-cell, the 120-cell and the 600-cell) and 6th, called the 24-cell, has no 3D analogue.

 From left to right, from top to bottom: Mary Stott, G.I. Taylor, Margaret Taylor, P.H. Schoute, A. Boole Stott [Boole Stott, undated(b)]. (Courtesy of the University of Bristol.)

She contributed to Hinton's 1888 book A New Era of Thought, about higher-dimensional reasoning, writing about sections of 3D solids and part of the preface. (By the time the book came out, Hinton and his wife Mary Ellen Boole Hinton had gone to Japan, following his conviction for bigamy for marrying a second wife under an assumed name. This surprising fact is perhaps less so when we recall he was the son of the radical proponent of polygamy, John Hinton). In 1889, she began secretarial work near Liverpool, working for lawyer and amateur mathematician H. John Falk, who was editing Hinton's book. In 1890 she met actuary Walter Stott, who joined Alicia and Falk on editing a new edition of Hinton's book. The pair married  had two children: Mary in 1891 and Leonard in 1892. According to Coxeter, "for some years she led a life of drudgery, rearing her two children on a very small income." In 1895 Walter told her of Dutch mathematician Pieter Schoute's work on central sections of the regular polytopes and she saw that his drawings matched her 3D models, so she mailed him photographs of her work. Schoute replied, asking to meet and collaborate which they did until he died two decades later. They wrote letters back and forth and some summers Schoute would come to stay with the Stott family in England so they could work together. He eventually convinced her to publish, which she did in two papers published in Amsterdam in 1900 and 1910. In her second paper, she was there first person to enumerate all 45 semi-regular polytopes. They also co-authored papers in 1908 and 1910. In 1907, they presented her models at the annual British Association of the Advancement of Science in Leicester. In 1912 Schoute presented his work on semiregular polytopes to the 5th International Congress of Mathematicians in Cambridge, crediting her with the roots of his proof. She made complete sets of models of 120-cell and 600-cell polytopes and left them with Schoute. When he died in 1913, Alicia took a hiatus from mathematics. The University of Groningen granted her an honorary doctorate in 1914, and exhibited her models. She was invited to the ceremony but not able to attend. When the degree arrived in the mail in a cardboard cylinder, she exclaimed, "This will be a good place to keep sticks of macaroni!" 

In 1930, when Alicia was 70, her nephew, physicist Geoffrey Ingram Taylor introduced her to the now famed British-Canadian mathematician Harold Scott MacDonald Coxeter who was 23 and a graduate student at Cambridge at the time. Despite their age difference they two became good friends. Like Taylor, Coxeter called her Aunt Alice. Coxeter wrote, "The strength and simplicity of her character combined with the diversity of her interests made her an inspiring friend." The two met regularly and began working together on topics in 4D geometry. He invited her to one of his supervisor H.F. Baker's celebrated geometry tea parties, where they presented a joint paper and she brought a set of her models, which she then donated to the department of mathematics. They collaborated on the investigation of a special kind of 4D polytope (Gosset's 4D polytope s{3,4,3}) for which she made models of its sections. Alicia made two more important discoveries about constructions for polyhedra related to the golden section. With Coxeter, she presented a joint paper at the University of Cambridge. When Coxeter moved to Toronto in 1936 to take up a professorship at U of T, Alicia gave him an antique stained-glass Archimedean solid shade and wrote to him, "My dear! I don't know how to write to you - words seem so futile besides so great a separation! But indeed one can rejoice, for your sake, that it happened so... While I have been writing my mind has gone back to the lovely world we have visited together, and which you have made so much your own. I wonder where you will get to in it! How I wish I could follow." They managed to continue their collaboration, despite their separation, until her death in 1940. Coxeter when on to work at U of T for 60 years and become regarded as one of the greatest geometers of the 20th century. Much of what we know about Alicia's life is thanks to Coxeter's reminiscing about his dear Aunt Alice.

My portrait contains her 3D unfolding of part of a hypercube, her colour drawings of some of the 3-principle sections of the 600-cell, her expansion of the octahedron - a truncated octahedron, sections of of the 16-cell.

p.s. I have related Alicia's life to some of her family, including mathematician father George Boole, mathematician, educator and activist mother Mary Everest Boole, brother-in-law mathematician (and bigamist) Charles Hinton, nephew physicist Geoffrey Ingram Taylor (who went on to be knighted and win the gold medal of the Royal Society, like his grandfather). But, I recommend reading Moira Chas' article 'The Extraordinary Case of the Boole Family' to also learn about further surprising and fascinating family. Alicia's sister Mary Ellen, first wife of Charles Hinton, was a poet. Charles Hinton also invented a baseball-pitching machine and wrote mathematical romances. Her sister Margaret (mother of Geoffrey Ingram Taylor) was a painter. Her sister Lucy Everest Boole was a Lecturer in chemistry at the London School of Medicine for Women and the first woman elected a Fellow of the Institute of Chemistry. Her sister Ethel was an activist concerned about the plight of the Russian people under Tsarist rule and travelled to Russian where she visited prisons, smuggled propaganda, composed music and wrote wildly successful novels (one of which was made into a film with music by Shostakovich). She married Polish exile and rare book expert Wilfred Voynich, who bought the famed, mysterious, medieval manuscript in an as-of-yet-undeciphered text illustrated with dreamlike images of strange plants and creatures, known as the Voynich manuscript. Alicia's son Leonard was a medical doctor and pioneer in the treatment of tuberculosis who inked a portable x-ray machine, an artificial pneumothorax apparatus and a system of navigation based on spherical geometry. Alicia's nephew Sebastian Hinton invented the jungle gym (inspired by the bamboo structures his father built while in self-imposed exile in Japan to teach his children about 3- and 4-D shapes). Sebastian married Carmelita Chase who founded the progressive Putney School, a boarding school in Vermont (where Coxeter was invited and considered working). Alicia's grandniblings Howard Everest Hinton became a distinguished entomologist, William Hinton a Marxist writer and Joan Hinton a nuclear physicist involved in the Manhattan Project who was so appalled by the consequences of the atomic bombing of Hiroshima she became a pacifist, gave up physics, moved to China and became a Maoist. Alicia's great-grandniblings include award-winning documentarian Carma Hinton and 2018 Turing Award-winning A.I. researcher Geoffrey Everest Hinton. 

References

Alicia Boole Stott, Wikipedia, accessed February, 2026. 

Boole Stott, Alicia. On Certain Series of Sections of the Regular Four Dimensional Hypersolids. Verhnadelingen Natuurkunde, Eerste Sectie, deel 7, nummer 3 (1900), pp. 1-21. 

Chas, Moira. The Extraordinary Case of the Boole Family. Notices of the American Mathematical Society. DOI: https://dx.doi.org/10.1090/noti1996. December, 2019. 

O'Connor, J.J. and E.F. Robertson, Alicia Boole Stott, MacTutor Archive, University of St Andrews. July 2014.

Polo-Blanco, Irene. Alicia Boole Stott, a geometer in higher dimension. Historia Mathematica. Volume 35, Issue 2. pp. 123-139. May, 2008.

Polo-Blanco, Irene. Alicia Boole Stott's models of sections of polytypes. Lettera Mathematica. Volume 2, pp. 149-154. September 9, 204.

Riddle, Larry. Alicia Boole Stott. Biographies of Women Mathematicians. Agnes Scott College. February 19, 2025

Mathematician Olga Ladyzhenskaya, Turbulent History, Fluid Dynamics and Partial Differential Equations

Olga Ladyzhenskaya, linocut print, 11" x 14" by Ele Willoughby, 2026

For the 6th #PrinterSolstice prompt 'expression' I chose Olga Ladyzhenskaya, a mathematician who overcame personal tragedy, faced Soviet totalitarianism and the catastrophic political upheaval of the 20th century to make a huge impact on the math, making important contributions to study of partial differential equations and left a legacy of influence on a school of students and collaborators in this field. She changed the very way partial differential equations are examined and was one of the leading figures to popularize the notion of weak solutions (which satisfy the equations but may not be differentiable) for partial differential equations. Expressions included in my portrait are for the Navier-Stokes equation, quoted directly from her 1958 paper on the subject*. She was known for her courage, kindness and integrity as well as her indelible mark on mathematics.

Born in the wild woods in Kologriv, a small, remote town on northwest of Moscow, Olga Ladyzhenskaya (1922-2004) and her older sisters were taught mathematics by their math teacher father Aleksandr Ivanovich Ladýzhenski. He would formulate a theorem and ask his daughters to prove it, a technique which fostered Olga's mathematical intuition. Olga showed the most promise amongst her sisters and soon she and her father were working on calculus together. His family had descended from nobility, and it was not a good time to be either an intellectual, or descendent from nobility, in the USSR. His uncle, who also lived with the family in Kologriv, was a famous watercolour painter, Gennardy Ladýzhenski. Olga was close with her great-uncle, whom she called "Dedushka" or grandfather, and she carefully preserved his landscape paintings of the River Ounja. Her mother Anna Mikhailovna was originally from a small town in Estonia and was the homemaker. Their home was full of books and the girls were exposed to culture, though they grew up in what felt like the hinterland. Olga had a lifelong love of art and literature. Aleksandr stood up for his students after their parents were arrested by the NKVD (the predecessor of the KGB secret police). In his epic The Gulag Archipelago Solzhenitsyn recounted that a peasant warned Aleksandr Ladýzhenski that he was on a list of enemies of the state, but he refused to hide. Shortly thereafter in 1937 he was arrested and executed, likely in a NKVD torture chamber like many other great teachers, declared an "enemy of the people." It was a great shock to his family. Olga's mother and eldest sister managed to support the family through this terrible time. Her mother made dresses, soap and other craft work to earn an income and survive. Olga's older sisters were expelled from school as daughters of Aleksandr. Her father's status, and the resulting impacts on his family, was not rehabilitated until 1956, when Khrushchev delivered the "Secret Speech" at the 20th Party Congress of the Communist Party which denounced the Stalin's purges and exonerated the teachers who had been killed. This was a speech which also finally allowed a resumption of communication between mathematicians, across the Iron Curtain. Prior to the speech the two groups were working in isolation of each other. Olga had been tackling the some of the most difficult equations of mathematical physics without the benefit of knowing about progress made in the West.

Despite her excellent grades, when Olga graduated from high school two years later, she was not admitted to Leningrad State University, as the daughter of an "enemy of the people." She was allowed to attend Pokrovski Teachers' Training College in Leningrad. Then during the war years, she was forced to leave Leningrad and she taught at an orphanage at Godorets before returning home to teach at the high school in Kologriv, like her father, as well as in her home. She welcomed all students, regardless of their ability to pay. In thanks for her kindness, a student's mother interceded on her behalf and Olga got an opportunity to study mathematics at Moscow State University. She was awarded a Stalin stipend (despite her family history) and ration card so she could survive as a student, but she was often hungry. Sometimes she had to sleep on the auditorium benches, joking that she was learning through osmosis by using her books as a pillow. 

She wrote a thesis supervised by renown mathematician Ivan Georgievich Petrovskiǐ.  She started studying algebra and number theory and her interest in partial differential equations (PDEs) grew. This is the calculus of multivariable functions, a way to determine the rate of change of a function with respect to one of its variables. This is a branch of mathematics which is invaluable to physics, engineering and other fields which use applied mathematics. It is often impossible to find explicit formulas for solutions for partial differential equations so a lot of modern math and science research goes into finding approximate numerical solutions for PDEs. In pure mathematics, there is research into what we can know qualitatively about the nature of solutions of certain important, but impossible to explicitly solve, PDEs: things like whether solutions exist at all, if so, whether are they unique, whether they are differentiable or smooth, whether solutions are regular or stable. 

After she graduated in 1947, she got married to Alekseevich Kiselev, a specialist in number theory and the history of mathematics who lived in Leningrad. Thus she moved to Leningrad for graduate school, where she taught in the physics department. Though they had a loving marriage, it was brief. She and her husband separated because he wanted children whereas she wanted to devote her life to mathematics and felt that children would be an obstacle. She remained single thereafter. 

Her official supervisor for her thesis on linear and quasilinear hyperbolic systems of partial differential equations was Sergei Sobolev, but unofficially she was guided by Vladimir Smirnov and they became friends. Smirnov was in charge of several branches of mathematics, seismology, hydrodynamics and aerodynamics and she was strongly influenced to study mathematical physics. She defended her PhD in 1951. Two years later, 1953 was an important year for Ladyzhenskya; She published her first book, she also defended her "habilitation" dissertation (the highest university degree awarded in some European nations typically 5 to 15 years after a PhD) and Stalin died, thus things slowly began to change in Russia. She went on to publish six monographs (some as long as 700 pages) and more than 250 papers! 

Ladyzhenskaya and Smirnov started the weekly mathematical physics "Smirnov Seminar," in 1947 and she took over the seminar after his death in 1974. She is remembered for asking just the right questions in seminars, which were revealing for teaching. She organized sporadic very popular conferences of differential equations and their applications.  She was known for her work on partial differential equations and especially her work on whether the solutions to regular problems in the calculus of variations are analytic, which is the 19th of famed mathematician David Hilbert's list of 23 (at the time) unsolved problems. In 1954 she joined the Stekov Institute mathematical physics laboratory and would go on to become its head in 1961. 

By the mid-50s she was working on problems in fluid dynamics, a particularly mathematically-challenging field of physics, and the Navier-Stokes equations for the motion of viscous fluids, in particular. These equations would interest her for the rest of her career. These partial differential equations are vital to modelling everything from the weather, to turbulence, to ocean currents, to flow in a pipe, or in blood vessels or around an airplane wing. Despite having been developed in the 19th century and their importance to such a wide array of science and engineering, to this day, there is much we still do not know about these equations. Turbulence may be common in everyday life but the physics and underlying mathematics remain some of the least understood. Physicists are interested in how the smooth laminar flow say in a river, breaks down when it hits an obstacle causing eddies upon eddies in the flow until we are left with turbulence - and whether we can correctly model that turbulence thereafter. In my portrait I show precisely this: laminar flow lines hitting an obstacle, the ensuing eddies and devolution into turbulence. Mathematicians however still have not even been able to show whether smooth solutions even always exist in three dimensions. We do not even know if the equations will allow us to model fluids with any given initial conditions, indefinitely into the future. This is called the Navier-Stokes existence and smoothness problem, deemed one of the seven most important unsolved math problems. The Clay Mathematics Institute is offering a 1 million US dollar Millennium Prize award for a proof or counter-example to the problem. 

In the 1930s, Jean Leray had demonstrated the existence of weak solutions of the Navier-Stokes equations, but it had proved more difficult to show whether solutions were unique, until Olga was able to show both existence and uniqueness. She did this at a time when, as a Soviet mathematician, she was not able to read Leray's work. She was the first to prove the convergence of finite difference methods to solve the Navier-Stokes equations. That means she showed that we can reliably find a solution to the equations of how viscous fluids move by using the very useful trick of approximating derivatives with finite differences. She also analyzed the regularity of solutions under certain conditions for two-dimensional flows. Her resulting monograph of the Navier-Stokes equations ranks amongst the most influential mathematical books ever published. She believed that when flow becomes highly turbulent in 3D systems that the Navier-Stokes equations are insufficient so she present her own modification to the equations at the International Congress of Mathematicians in Moscow in 1966. 

She also analyzed the regularity of other types of partial differential equations, including parabolic equations (Vsevolod A. Solonnikov and her student Nina Ural'tseva) and quasilinear elliptical equations. These two types of equations are invaluable to the physics and engineering of time-dependent and steady-state phenomena, respectively. In the 1960s, these three published a veritable encyclopedia on the subject of the regularity of the solutions of PDEs that remains authoritative today. She published her influential text The Mathematical Theory of Viscous Incompressible Flow in 1961. She dedicated the book to the three people she most respected: her father, Vladimir Ivanovich Smirnov and Jean Leray.  At a time when exchanges between Western and Soviet scientists and mathematicians were virtually non-existent, she extended the results of Ennio De Giorgi, Jurgen Moser and John Nash ( the 1994 Nobel Laureate in Economics). 

Olga Ladyzhenskaya

She taught mathematics throughout her career and was recognized as one of the most significant and influential 20th century mathematicians. She nominated for the 1958 Fields Medal. She was president of the St Petersburg Mathematical Society. She was awarded the 1969 State Prize of the USSR and the Chebyshev Prize of the USSR Academy of Science. She was elected a corresponding member of the USSR Academy of Science in 1981, a foreign member of The German Academy of Scientists Leopoldina in 1985, a foreign member of the Academia die Lincei in 1989, a full member of the Russian Academy of Sciences in 1990 and a foreign member of the American Academy of Arts and Science in 2001. In 1989 Communist rule in the USSR ended, and Olga could travel more freely. She won the Great Gold Lomonosov Medal in 2002. Her name is inscribed in marble on a table at the Science Museum of Boston alongside the other most influential 20th century mathematicians. 

She counted amongst her friends dissent writer Aleksandr Solzhenitsyn, famous for raising awareness of repression and the Gulag prison system in Soviet Russia, and poet Anna Akhmatova, who wrote about the Stalinist terror, was twice nominated for the Nobel Prize for literature and dedicated a poem to her. She loved nature and travelling and was a skilled storyteller. She was once on the city council, and was engaged in her community, often risking her own safety and career to aid people opposed to the Soviet regime. She helped many mathematicians in Leningrad obtain apartments, free of charge, for themselves and their families. She cared deeply, was beloved and known to be full of energy and as a person of integrity, courage, faith, and unafraid to express her viewpoint despite a dangerous political climate. Nonetheless she remained a patriotic Russian and encouraged fellow Russian mathematicians to remain in Russia. She was plagued by eye problems and relied on special pencils to work in later years. She loved the sun and with her vision loss found the dark St Petersburg winters a challenge in later life. She was about to make a trip to Florida, and complete a paper on computational hydrodynamics when she died in her sleep in 2004 at the age of 81.

*The text reads "систему уравнений Навье — Стокса" which means 'a system of Navier-Stokes equations' and also "для функций" which means 'for functions' and "при граничных и начальных

условиях" which means 'under boundary and initial conditions'.

References

Dumbaugh, Della, Panagiota Daskalopoulos, Anatoly Vershik, Lev Kapitanski, Nicolai Reshetikhin, Darya Apushkinskaya, and Alecander Nazarov, The Ties That Bind - Olga Ladyzhenskaya and the 2022 ICM in St. Petersburgh. Notices of the American Mathematical Society. DOI: https://dx.doi.org/10.1090/noti2047. March, 20220.

Friedlander, Susan, Peter Lax, Cathleen Morawetz, Louis Nirenberg, Gregory Seregin, Nina Ural'tseva, and Mark Vishik. Olga Alexandrovna Ladyzhenskaya (1922-2004). Notices of the AMS, Volume 51, Number 11, pp. 1321-1331, December, 2004. 

Goudon, Thierry and Irina Sophia Antipolis. Olga Alexandrovna Ladyzhenskaya. Brèves de Maths - Mathématiques de la planète Terre. April 23, 2013.

O'Connor, J.J. and E.F. Robertson, Olga Alexandrovna Ladyzhenskaya. MacTutor. University of St. Andrews, August, 2005.

Olga Ladyzhenskaya. Wikipedia, accessed January, 2026.

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

Virginia Ragsdale's Conjecture

Virginia Ragsdale, linocut, 11" x 14" on cream coloured washi, by Ele Willoughby, 2022

One of the unexpected delights of making a series of portraits of people in STEM is that people are interested in subjects because of their personal relationships. In hindsight, it should have been obvious; I am a scientist and have family and friends! But, it didn't occur to me that these "famous" (in some circle) great scientists would appeal as images of family members, or friends, and it's actually pretty touching when people contact me and tell me they are buying a print because the subject is someone they knew personally. This is a mathematician who was introduced to me by a repeat customer, because this is one of her ancestors. So not only to I get to learn of a new-to-me historic women in STEM, but I have a commission to make her portrait for someone's own personal connection to her.

This is a linocut portrait of mathematician Virginia Ragsdale (1870-1945) who is remembered for the Ragsdale conjecture (shown in the upper two inequalities). I'm always wary of including equations in talking about the history of math and science. What's important is that she proposed a way of approaching a difficult problem that was an inspiration for a lot of other mathematicians. The problem (which remains unsolved!) was included in David Hilbert's famous set of problems, namely, what are the possible arrangements of real algebraic curves embedded in the projective plane? She decided to set an upper bound for certain type of such curves. She proposed that they consider algebraic curve of degree 2k which are all topologically circles (or ovals) where some ovals are nested inside each other; others are not. An oval is defined as even if it is contained an an even number of other ovals of the curve, otherwise the oval is called odd. She conjectured that for an algebraic curve which contains p odd and n even ovals:

 

and

.

It was a very useful insight to consider even and odd ovals separately; the difference p-n is the Euler characteristic of a region bounded by the even and odd ovals. The Ragsdale conjecture, made in her 1906 dissertation, is amongst the earliest and most famous on the topology of real and algebraic curves, which stimulated a lot of 20th century research and was not disproved until 1979. A correct upper bound has yet to be found. I have also included an inequality she posed:

later proved by Ivan Petrovsky. The diagrams of algebraic curves also appeared in her dissertation "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. In it she tackles the 16th of David Hilbert’s famous 23 unsolved problems in mathematics - one of only a few which remain unresolved today. 

Born in Jamestown, North Carolina just after the Civil War, she was class valedictorian at the Salem Academy, where she excelled at math and piano. She was went on to Guilford College where she helped set up a YMCA, and establish an Alumni Association as well and worked to expand college athletics. When she graduated with a B.S. in 1892, she won a scholarship to Bryn Mawr for the woman student with the highest grades, so she continued on, studying for her A.B. degree in physics. She won a fellowship to study in Europe for the class of 1896, which she delayed for a year, working as a physics demonstrator and beginning graduate studies in math. She then spent 1897-98 attending lectures by the renown mathematicians Felix Klein and David Hilbert at the University of Göttingen.

Upon returning the the US, she taught math for three years in Baltimore before another scholarship, awarded by the Baltimore Association for the Promotion of University Education of Women, allowed her to return to Bryn Mawr to complete her doctorate with Charlotte Scott. 

She moved to New York and taught at Dr. Sach’s School for Girls until 1905. She became the head of the Baldwin School in Bryn Mawr from 1906-1911 and worked as Charlotte Scott’s reader from 1908-1910. She accepted a mathematics position which brought her back to North Carolina in 1911 at the Women’s College in Greensboro (now UNC at Greensboro) where she stayed for almost two decades. She was department head from 1926-1928 and left a lasting impact, insisting on investing in a telescope and adding statistics to the curriculum. She held high standards but was known for her patience for students.

She retired to care for her ailing mother and run the family farm in 1928. When her mother died she built a lovely house at the edge of the Guilford College where she gardened, restored furniture and researched her family’s genealogy. Upon her death she left the home to the college and serves now as the home of the college president. 

References

Virginia Ragsdale, On the Arrangement of the Real Branches of Plane Algebraic Curves, American Journal of Mathematics, Volume 28, 1906

Virginia Ragsdale, Wikipedia, accessed April 2022

Ragsdale conjecture, Wikipedia, accessed April 2022

De Loera, Jesús; Wicklin, Frederick J. "Biographies of Women in Mathematics: Virginia Ragsdale". Anges Scott College. Retrieved April, 2022.

Poincaré, Cubism, Non-Euclidian Geometry and Special Relativity

Cubist Poincaré, linocut, 8" x 8" by Ele Willoughby, 2022

This is a hand-printed lino block portrait in the Cubist style of mathematician Henri Poincaré. It’s one of a variable edition printed on Japanese kozo (or mulberry paper), 8” x 8” or (20.3 cm x 20.3 cm). The latest prompt for #printersolstice is Cubism.

Diagram from Jouffret's 'Traité élémentaire de géométrie à quatre dimensions'

This print is about how movements in art can be connected with contemporary math and physics (and other sciences). Specifically, the way Cubism breaks from a single favoured perspective or absolute frame of reference and attempts to break down subjects into geometrical shapes from multiple points of view can be tied to advancements in non-Euclidian geometry in mathematics, special relativity (published by Einstein in 1905) and the revolution in early 20th century physics. Picasso and his circle were specifically introduced to the mathematics of the fourth dimension and Non-Euclidian geometry by his actuary and enthusiastic amateur mathematician friend Maurice Princet, who gave him the book 'Traité élémentaire de géométrie à quatre dimensions' (Elementary Treatise on the Geometry of Four Dimensions) by Esprit Jouffret, a popularization of Henri Poincaré's 'La Science et l'Hypothèse' (Science and Hypothesis). Jouffret's book includes many diagrams of four dimensional objects like hypercubes projected on the 2D surface of a page. Princet also spoke to Picasso about the work of polymath Henri Poincaré. Picasso realized that this approach to 4D geometry captured the how he wanted to depict two points of view simultaneously and art historians can trace the influence of this geometry in his earliest Cubist paintings. So I have made a Cubist style portrait of mathematician, physicist, engineer and philosopher of science Henri Poincaré (1854-1912).

Henri Poincaré 
(photograph published in 1913, via Wikipedia)

Pointcaré was called "The Last Universalist" since he excelled in such a wide range of mathematics including pure and applied math, mathematical physics and celestial mechanics. Chaos theory and the idea that deterministic systems can have sensitive dependance on initial conditions (colloquially known as the butterfly effect) seemed to explode on the scene with increasing applications in the 1980s but in fact some foundations were laid a century earlier by Poincaré. He was the first person to discover a deterministic system which exhibits chaos in the three-body problem (or how a system of  3 masses behave over under Newton's Law of Universal Gravitation). The three circular shapes in the background allude to the 3-body problem and the spiral is a hint at the chaos in the trajectories. He is considered a founder of the study of topology (how geometric properties are preserved under deformations) and stressed the importance of the invariance of physical laws under various transformations. Special relativity tells us that light speed is the universal speed limit, and that space and time are not absolute. As things approach this speed limit they experience length contraction and time dilation. We use Lorentz transformations to compare length and duration in differing frames of reference. It was Poincaré who wrote the Lorentz transformations in their modern form. He discovered the Lorentz transformations for velocity and the invariance of Maxwell's equations (for electricity, magnetism and light), which was vital part of special relativity. He published a short paper in 1905 which Einstein did not see until after he had published his own independent and revolutionary paper on special relativity. Poincaré also proposed gravitational waves, travelling at light speed from masses. It's hard to over-state his importance to 20th century math and physics.

Pablo Picasso, Portrait of Ambroise Vollard, spring of 1910 (Pushkin Museum, Moscow)

Poincaré himself worked quite intuitively and wrote, "It is only through science and art that civilization is of value."

One of the ways in which Special relativity is strange is that we discovered the problem of simultaneity. If there is no absolute frame of reference, there is no simultaneity; observers in different inertial frames of reference (that is going different speeds*) will not agree on the sequence of events or even whether two things happen at the same time. This problem of simultaneity, it can be argued, is exactly what interested Picasso, when he chose to paint two different points of view on the same canvas. This is the argument made by Miller, author of 'Einstein, Picasso: Space, Time and the Beauty That Causes Havoc' makes. Leonard Schlain, in 'Art & Physics' makes the case that Picasso in painting all sides of an object simultaneously as if he was seeing space as all here in the everlasting now, was depicting what a viewer would see astride a beam of light at the universal speed limit when time has reached its maximum dilation to the present (and only the present). 

Earlier art historians, and in fact Einstein when asked, argued that Cubism wasn't connected to relativity or non-Euclidian geometry but I think each under appreciated the other. The art historians argued the artists had no knowledge of Einstein, or Minkowski (who combined space and time into the 4D spacetime), equations or pure math. Einstein preferred earlier classical painting and argued "This new artistic 'language' has nothing in common with the Theory of Relativity" but then, he felt contemporary art and music had both degenerated. If we are less literal, I think the connection between the two movements is quite clear. There is no absolute frame of reference and both art and physics were revolutionized by this realization.

Schlain makes a further argument connecting Cubism with relativity but noting that colours also change with speed, as length contracts (including the wavelengths or colours we would observe), and colours would be merge as what is in front and behind become one. He argues we would be left with neutrals at the speed of light: white (of white light), black (of its absence),  and muddy tones of mixed colours brown and grey. These are precisely the colours Picasso and Braque used in their early Cubist paintings, and the colours of my palette for my portrait of Poincaré.

*without acceleration and/or far from large masses, so we can ignore general relativity

References

William B. Ashworth, Jr., Scientist of the Day - Esprit Jouffret, Linda Hall Library blog, March 15, 2021.

Arthur I. Miller, Henri Poincaré the unlikely like between Einstein and Picasso, The Guardian, July 17, 2021

Did Picasso know about Einstein?, Physics World, November 1, 2002

Leonard Schlain, Art & Physics, Quill William Morrow, New York, 1991

Henri Poincaré, Wikipedia, accessed January, 2022

Mathematician Karen Uhlenbeck, Bubbles and Bubbling

Karen Keskulla Uhlenbeck, linocut detail, 9.25" x 12.5", variable edition on Japanese kozo paper with chine collé with suminagashi marbling, by Ele Willoughby, 2021

My latest #sciart (or, maybe #mathart) portrait is of mathematician Karen Uhlenbeck. Though a lot of her work is pertinent to physics, this was still a challenge to work out how to tell the story of her mathematics. A founder of modern geometric analysis, she won the 2019 Abel Prize for "her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." Partial differential equations are pretty central to doing physics, and a sort of mathematics I really enjoy, but apart from that, I know what all the words mean in that previous sentence, but things like gauge theory and analysis aren't part of the math I've studied or used in my own research. As we move forward in time, depicting contemporary (or near contemporary) mathematics is increasingly challenging because it's beyond my ken.

I looked up several of her important papers, seeking geometrical diagrams as a way to tell the story of her research pictorially. Though she describes herself as a visual, geometric (and messy!) thinker, I didn't find anything but a lot of algebra (which I would have had to work hard to follow, let alone use to speak to a wide audience, or painstakingly carve in reverse!) But I found an "in" for depicting minimal surfaces and her methods for removing singularities, through Fields Medalist Sir Simon Donaldson's review of her work, including a schematic of "bubbling" and also some of the great science/math communication and articles about her and her work, which were published when she won the Abel Prize.

Born in Clevland to an engineer father Arnold Keskulla and teacher and artist mother Carolyn Windeler Keskulla, her family moved to New Jersey when she was a child. She spent her childhood enjoying nature, playing football and secretly reading science texts, imagining a job where she could avoid people. She started a degree in physics but switched to math at the University of Michigan. In the wake of the Soviet Union's launch of Sputnik the US Congress passed the National Defense Education Act, to encourage more participation in STEM including encouraging women and minorities in those fields. The university added an honours math course in response, which showed her there was more to math than she ever knew and it changed her life. “The structure, elegance and beauty of mathematics struck me immediately, and I lost my heart to it,” she wrote. She switched majors. She chose the Courant Institute of Mathematical Sciences at New York University, for graduate school, avoiding applying to more prestigious schools where she feared the competitive male math culture wouldn't suit her. She married biophysicist Olke C. Uhlenbeck, the son of European intellectuals; his father is well-known physicist George Uhlenbeck. She was quite influenced by her new in-laws. Her parents valued ideas but put an emphasis on earning an income, but her in-laws valued intellectual things and encouraged her pursuit of mathematics. Her husband moved to Harvard and she opted to switch to graduate school at Brandeis University, where she earned her PhD in 1968 with advisor Richard Palais. Palais was exploring the space between analysis (a generalization of calculus) and topology (the study of shapes) and Uhlenbeck enjoyed this in-between world. With Smale, Palais had made advances in the study of harmonic maps, which came out of the calculus of variations and the search for shapes in equilibrium with physical measurements. Uhlenbeck's research built on this.

She found temporary jobs at MIT and Berkeley but was hampered by the classic "two-body problem" for academic spouses (where an couple cannot find two professorships in the same place) and anti-nepotism rules - though she later noted that the institutions which claimed they were precluded from hiring both her and her husband, even in different departments, did not officially have any such anti-nepotism rules on the books. Schools like MIT, Stanford and Princeton which courted her husband, simply declined to hire a woman in math. Her husband generously declined to work at institutions where she could not.

Eventually they both got faculty positions at the University of Illinois at Urbana–Champaign in 1971. But she could not stand life in Urbana, where she was often dismissed as an "academic wife" rather than a professor in her own right. She lacked professional support, did not enjoy teaching, felt undervalued and struggled to find the right research topic for her. But, while there, she met post-doc Jonathan Sacks and they began working whether certain problems converge to harmonic maps; eventually they discovered a sort of "awkward point" or singularity called a bubble where systems cannot converge on a harmonic map. The mathematical concept of a minimal surface minimizes area (or surface tension or energy) the way a soap bubble does in the natural world. They showed there were a finite number of these singularities and that when you rescale a map near them you have "bubbling" as in the diagram in my print. This trick of rescaling was one of her and Sacks' most important contributions.

 She moved to the University of Illinois at Chicago in 1976, separating from her husband. In 1983, she moved to the University of Chicago, where there were other women math professors and peers who took her seriously. She received a Sloan Fellowship and finally knew she could support herself and her research. She learned the value of a professional support network and realized that isolating herself was hurting her and her career. Her work with Sacks was foundational to the development of the new mathematical field of geometrical analysis. This bubbling phenomenon was found in many different sorts of mathematical physics. She took her intuition and techniques honed working on minimal surfaces and began ground-breaking work in gauge theory and the nature of exotic n-spaces, a means of connecting observations at different places that comes up in the quantum mechanical world of fundamental forces, building on the work of mathematical physicists like Weyl, Yang and Mills. With C.H. Taubes, she analyzed Yang-Mills equations in four dimensions; she proved her removable singularities theorem, showing that bubbling cannot occur around isolated points, so if a finite-energy solution exists to the Yang-Mills equations near a point, the solution will extend smoothly to the point. Simon Donaldson's extension of this research won him the prestigious Fields Medal for mathematics in 1986. Uhlenberck won a MacArthur Prize Fellowship or "genius grant" (1983-1988) which made her feel, "obliged to become more ambitious.”

She married mathematician Robert F. Williams, then in 1988, moved to the University of Austin as the Sid W. Richardson Foundation Regents Chairholder and won the Noether Lecture award from the Association for Women in Mathematics. In 1990, she was only the second woman in math to give the plenary lecture at the International Congress of Mathematicians in Kyoto, after none other than mathematical giant Emmy Noether. She founded the Park City Mathematics institute and with Chuu-Lian Terng she co-founded the IAS Women and Mathematics (WAM) program to recruit and retain more women in math, encouraging collaboration and mentorship at all stages of their careers, a complete turn-around from the isolation she sought at the beginning of her career and her avoidance of her fellow women in math because “It was self-evident that you wouldn't get ahead in mathematics if you hung around with women.” Together, they've created a network of nearly 1500 women mathematicians. She realized that one needs role models - not examples of perfection, but people who make errors but pick themselves back up. When asked if she had a role model, she cited beloved chef and TV presenter Julia Child! "She had these fantastic television programs, and she was a real person. She could pick the turkey up off the floor and serve it," Uhlenbeck said. She founded the Distinguished Women in Mathematics Lecture Series at the University of Texas at Austin. Uhlenbeck won the National Medal of Science in 2001 and the 2007 Steele Prize for a Seminal Contribution to Mathematical Research from the American Mathematical Society. Granted yearly by the King of Norway, rather like a math equivalent to the Nobel, the Abel Prize includes a $700,000 award, and recognizes a mathematician’s entire career. Uhlenbeck became the first (and so far only) woman to win the Abel Prize in 2019.  She donated half the funds to organizations encouraging underrepresented (that is, traditionally excluded) minorities in mathematics. She further donated to environmental organizations and the National Academy of Science. She is now an emeritus professor at Austin, as well as a visiting associate at the Institute for Advanced Study and senior research scholar at Princeton University.

Like so many of my subjects, I've found she also loves art and began drawing more than a decade ago, mostly landscapes. She notes, “I discovered the fascinating fact that the problem of scale occurs both in mathematics and in drawing.” In drawing you want to capture the full forest as well as the details of individual plants, just as you want to capture all scales in math. “The hardest part with both is fitting the two scales together. You need the right tools.” Uhlenbeck has provided the world of mathematics and physics with so many of just the right tools.

References

Donaldson, S.. “Karen Uhlenbeck and the Calculus of Variations.” Notices of the American Mathematical Society 66 (2019): 1. DOI:10.1090/noti1806

Sioban Roberts, "In Bubbles, Karen Uhlenbeck Sees a Mathematical Universe," The New York Times, April 8, 2019

Karen Uhlenbeck, Wikipedia, accessed July 2021.

Rachel Crowell, "Karen Uhlenbeck Becomes First Woman To Win Abel Prize For Mathematics," Forbes, March 20, 2019

Dale DeBakcsy, How Mathematical Lone Wolf Karen Uhlenbeck Found Her Pack, Women You Should Know, November 29, 2017.

Karen Uhlenbeck, Karen Uhlenbeck on Being the First Woman to Receive the Abel Prize, remarks made at reception at the Institute for Advanced Study on March 19, 2019.

Karen Uhlenbeck, Coming to grips with success: a profile of Karen Uhlenbeck, Celebratio Mathematica, 1996

Jessica Atlee, Crossing Fields: Karen Uhlenbeck’s pioneering work marries math with physics, Symmetry Magazine, August 15, 2019.

Erica Karreich, Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize, Quanta Magazine, March 19, 2019

Marianne Freiberger, The Abel Prize 2019, Plus magazine, University of Cambridge, March 19, 2019.