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

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.

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.

Mathematician Sophie Germain

 

Linocut 'Sophie Germain' by Ele Willoughby, 2021

French mathematician, physicist and philosopher Marie-Sophie Germain (1776 – 1831), known as Sophie, taught herself mathematics using books in her father's library and by corresponding with leading mathematicians of her day, including Lagrange, Legendre and Gauss, initially using the pseudonym Monsieur LeBlanc since she knew it was unlikely her contemporaries would take a woman in math seriously. She later wrote to Gauss, the leading mathematician of her day that she used the pseudonym "fearing the ridicule attached to a female scientist." Her work on elasticity theory won her the grand prize from the Paris Academy of Sciences. Her trailblazing work on Fermat's Last Theorem set out a strategy and framework for mathematicians pursuing the problem for hundreds of years. She was denied a formal mathematics education or professional standing due to her sex, but Gauss argued she deserved an honourary degree (though it was never granted).

 

She was born to a bourgeois family; her father is usually described as a wealthy silk merchant who was elected to the États-Généraux (later the Constitutional Assembly), as a representative of the bourgeoisie. By the time she was 13, the Bastille fell, and she remained indoors for safety during those tumultuous times, her father's library her only source of entertainment. She was fascinated reading about Archimedes' death at Roman soldiers' hands during the siege of Siracusa, unable to tear himself away from mathematics. It piqued her interest and she devoured the mathematics books she found, even teaching herself Latin and Greek in order to read Sir Isaac Newton and Leonard Euler's works. Her parents disapproved this sudden interest in a subject deemed inappropriate for a woman and they denied her warm clothes or a fire to study at night. She simply wrapped herself in quilts and studied by candlelight, when it was so cold her ink froze, eventually winning over her mother's support. Sophie was lucky in that she was not forced to marry and her family was wealthy enough to support her throughout her life, and though mostly excluded from formal education and the society of mathematicians, she was able to pursue her self-study.

 

The École Polytechnique opened - to men only - in 1794, but lecture notes were made available on request. Students were requested to send solutions to faculty and Sophie began submitting notes to Joseph Louis Lagrange under the name of a former student who had died young, Monsieur Antoine-Auguste Le Blanc. Lagrange recognized "Le Blanc's" ability and requested a meeting, so her rouse was up. Luckily, Lagrange was not opposed to a woman studying mathematics and agreed to mentor her. In 1798, Adrien-Marie Legendre published Essai sur la théorie des nombres and Sophie became interested in number theory and began corresponding with Legendre. Impressed, he included some of her work in the supplement to the second edition of his book, praising it as très ingénieuse ("very ingenious"). Then Gauss published his magnum opus Disquisitiones Arithmeticae. She worked through it for three years before writing him, again as M. Le Blanc, to discuss his book and tell him about her work on Fermat's Last Theorem, a famous grand problem of number theory. Unfortunately, she had made a weak assumption in one of her proofs, and Gauss did not reply to this first letter.

 

Mathematician Pierre de Fermat famously scrawled his eponymous last theorem in the margin of a book around 1637, without supplying any proof, noting simply that the proof was too long to fit. After 354 years of effort by countless mathematicians, Andrew Wiles was finally able to prove the theorem correct in 1995. The theorem states that no three positive integers x, y, and z satisfy the equation x^p + y^p = z^p for any integer value of p greater than 2. Sophie was working on this problem and making real in-roads.

 

During the Napoleonic wars, France occupied the German town of Braunschweig, where Gauss lived, and Sophie feared he might suffer the same fate as Archimedes. She wrote family friend General Pernety pleading for him to ensure Gauss' safety. Soldiers were dispatched and found the confused Gauss perfectly safe. Gauss, of course, did not know that Sophie Germain was none other than M. Le Blanc. She decided to reveal her identity and he replied,

 

How can I describe my astonishment and admiration on seeing my esteemed correspondent M. Le Blanc metamorphosed into this celebrated person ... when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the noblest courage, extraordinary talent, and superior genius.

 

They became friends and Gauss truly respected her ability, though he was not a reliable correspondent and generally did not review her work (and she would have really benefited from such feedback, lacking mentorship in number theory, and having gaps in her knowledge since she was self-taught). 

 

She became interested in other problems. German physicist and musician Ernst Chladni had published his experiments on vibrating plates (following the trailblazing work of Robert Hooke). He used a violin bow to vibrate a metal plate covered in sand, so that the sand would concentrate on nodal lines marking divisions between regions that moved in opposing directions. His drawings of the patterns produced are known as Chladni figures (like those shown in lavender in my print). Germain was able to attend his demonstration in Paris. The Paris Academy of Sciences launched a contest to develop the mathematics explaining the vibration of an elastic surface and comparing this to experimental data like Chladni's, with a reward of 3,000 francs. Lagrange pointed out that a new branch of analysis would be required and scared off all would-be contestants with the exception of Sophie and Denis Poisson. But Poisson was elected to the Academy, thus became a judge, leaving only Sophie. She began, mentored by Legendre, but her submission was deemed insufficient, though she provided some ingenious results, which allowed Lagrange to derive an equation, correct under certain conditions. Lagrange died within two years, and Sophie lost a mentor. The Academy decided to extend the contest and Sophie persisted. After initially helping, Legendre withdrew his support. Sophie submitted another attempt anonymously, but it had several errors (of the sort she would have been taught to avoid had she been allowed to study math at a university). She consulted Poisson, and he had access to all her notes as a judge. He then published his own work on elasticity without acknowledging any of her work or their conversations on the subject. At that point in 1816, she published under her own name, "Recherches sur la théorie des surfaces élastiques" partially so what Poisson had done would be clear, and to point out the errors in his work. They extended the contest again partially in response to the breach of confidentiality by Poisson and she persisted with her efforts. She won the gold medal and became the first woman to win a prize from the Paris Academy of Sciences but did not attend the prize ceremony. The Academy was not entirely satisfied; she had the correct differential equation, but having used an incorrect equation by Euler she had incorrect boundary conditions. Even as a prize winner, she was still denied entry to academy meetings as a woman for several years until she made friends with Joseph Fourier, a secretary of the Academy, who got tickets on her behalf. She published her prize-winning essay in 1821, at her own expense as the Academy had neglected to do so, complete with her notes on errors she had made. In 1826, she submitted a revised version of her work; the Academy considered it trivial but they did not want to reject her as they would a man and professional colleague. They both denied her access and were patronizing in their misguided attempt at kindness. She published this essay on the advise of mathematician Augustin-Louis Cauchy. Her nephew later made sure she had a final publication on elasticity, publishing "Mémoire sur la courbure des surfaces" posthumously on her behalf in 1831.

 

In 1815, the Academy offered an award for a solution to Fermat's Last Theorem, rekindling her first love of number theory. She wrote Gauss with her strategy for a general proof and the significant in-roads towards a proof she had made, but Gauss never replied. She produced what is now known as Sophie Germain's Theorem. To show that Fermat's Last Theorem holds, you can divide the powers p into numbers which are not divisors of x, y or z, or powers p which are a divisor of at least one of x, y or z. She proposed her theorem:

 

Let p be an odd prime. If there exists an auxiliary prime P = 2Np + 1 (N is any positive integer not divisible by 3) such that:

1. if x^p + y^p + z^p ≡ 0 (mod P), then P divides xyz, and
2. p is not a p-th power residue (mod P).

and she used this result to show that Fermat's Last Theorem holds true for all odd primes p < 100. Her method was later used to show it holds true for all p < 1700. Her theorem is known from a footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5. The text in my print is from one of her on unpublished manuscripts:

Remarque sur l'impossibilité de satisfaire en nombres entiers a l'équation x^p + y^p = z^p. L'impossibilité de cette équation serait hors de doute si on pouvais démontrer la théorème suivant: Pour toute autre valeur de p que p = 2, il y a toujours un infinité de nombres premiers de la forme Np + 1 pour lequels on ne peut trouver deux residus 1^ièmes puissances  dont la différence soit l'unité.

 

(My gloss: "Remarks on the impossibility of any whole numbers satisfying x^p + y^p = z^p . The impossibility of this equation can be shown to be without doubt if we can demonstrate the following theorem: For all p > 2, there are an infinite series of primes of the form Np + 1 for which we cannot find two residues of the first power separated by 1"). She goes on to note that if there's any numbers which do satisfy Fermat's equation for p > 5 it must be numbers "whose size frightens the imagination", around 40 digits long.  

 

She also pursued philosophy and psychology on her own. Her nephew had two of her works published posthumously: Pensées diverses, a history of science and math with her commentary and Considérations générales sur l'état des sciences et des lettres, aux différentes époques de leur culture in which she argued there no difference between the sciences and the humanities and gained the praise of philosopher August Comte. 

 

She continued working despite pain after her diagnosis of breast cancer in 1829. She died in 1831, listed only as a property owner, not a mathematician on her death certificate. She is now recognized for her brilliance and originality, but her progress was often sadly hampered by the lack of instruction and the way her peers treated her as a novelty, and avoided proper constructive criticism. She wrote, “These facts are my domain and it is to me alone that they remain hidden. That’s the privilege of the ladies: they get compliments and no real benefits.” Several scholars argue that a contemporary man with similar skills and interest would have had his abilities embraced and talents nurtured. Sophie Germain was able to achieve what she did through both her tremendous talent and extraordinary persistence. Along with her theorem, subsequent discoveries in number theory have been named in her honour, as well as a street in Paris and the Sophie Germain Prize in mathematics offered by the same Academy which had snubbed her.

 

References

Sophie Germain, Wikipedia, accessed March 2021

Sophie Germain’s Theorem, Wikipedia, accessed March 2021

Ernst Chladni, Wikipedia, accessed March 2021

Ernst Chladni, Entdeckungen über die Theorie des Klanges, 1787, via Chladni Figures (1787) on Public Domain Review

Cristina P. Tanzi, Sophie Germain's Early Contribution to the Elasticity Theory, MRS Bulletin , Volume 24 , Issue 11 , November 1999 , pp. 70 - 71 DOI: https://doi.org/10.1557/S0883769400053549

Alexanderson, G.. “About the cover: Sophie Germain and a problem in number theory.” Bulletin of the American Mathematical Society 49 (2012): 327-331.

Richard Baguley, Sophie Germain: The Mathematics Of Elasticity, Hackaday.com, March 20, 2018

Evelyn Lamb, Thank You, Sophie, and I'm Sorry, Scientific American Blog, April 1, 2017.

Maria Popova, How the French Mathematician Sophie Germain Paved the Way for Women in Science and Endeavored to Save Gauss’s Life, Brainpickings, org, February, 2017.

Reinhard Laubenbacher and David Pengelley,  “Voici ce que j’ai trouvé:” Sophie Germain’s grand plan to prove Fermat’s Last Theorem, Historia Mathematica Volume 37, Issue 4, November 2010, Pages 641-692