Elisabeth Koopmann-Hevelius Set Her Sight on Being an Astronomer

Elisabeth Koopmann-Hevelius, linocut print 11" x 14" by Ele Willoughby, 2026

For the 11th #PrinterSolstice2526 prompt angle, I've made a portrait of astronomer Elisabeth Koopman Hevelius using (sextant or ) an octant, literally measuring angles to astronomical bodies. 

One of the earliest recorded women astronomers, Elisabeth Catherina Koopmann-Hevelius (Elżbieta Heweliusz, 1647-1693) was born to a wealthy, land-holding, Dutch Lutheran, merchant Nicolas Koopman (1601-1672) and his wife Joanna Mennings (or Mennix; 1602-1676) in the largely German-speaking city of Danzig, then part of the Pomeranian Vovoidship of Royal Prussia in the Polish-Lithuanian Commonwealth, part of the Hanseatic League, now known as Gdańsk, Poland.  Her parents had been married in Amsterdam in 1633, then moved to Hamburg and again to Danzig by 1636. Her education included languages (including at some point Latin, the international language of science of the day) and natural sciences. Elisabeth was fascinated with astronomy from the time she was a child. The year she was born, local politician, brewer and astronomer Johannes Hevelius (1611-1687) published his beautifully and elaborately illustrated Selanographia, about the moon, the first geographical book about a body other than the Earth. By 1650 he already had an international reputation, a complex of three houses and 200 square metre observatory equipped with several large telescopes, the largest observatory in Europe. The young Elisabeth approached him and he promised he would show her the splendour of the night sky when she was older. In 1662, his first wife Katharina Rebeschke died. Elisabeth had become ever more fascinated by astronomy and had realized that Johannes was in fact a renown astronomer, and her admiration of him grew. She reminded the elder astronomer of his promise. Despite their age difference, based largely on their mutual love of astronomy, the two decided they could be happy together and wed when Elisabeth was 16 and he was 52, in 1663. Such an age gap would not have shocked their contemporaries, and for a young woman like Elisabeth, barred from university education, marriage would have been the only means for her to pursue astronomy. Elisabeth had to run the complicated Hevelius household and she both assisted Johannes and pursued her own astronomical interests. They had four children; a son, who died in infancy and three surviving daughters. 

Detail of Johannes and Elisabeth Hevelius observing the sky with a brass octant (1673).

While awaiting the arrival a new assistant, Elisabeth aided her husband in his observations; she excelled at the job and loved observing. In 1663, France awarded him a pension for his astronomical work, and Johannes began corresponding with Henry Oldenburg, secretary of the Royal Society in London. Elisabeth became his partner in observation from 1664 onwards. That year, Johannes was inducted as the first foreign member of the Royal Society of London. Also that year their son John Adeodatus was born but he died a year later. Their daughters Catherine Elisabeth, Julia Renata and Flora Constance were born in 1666, 1668 and 1672, respectively.  In 1668, Johannes published their work in Cometographia, listing comets and sunspots. Showing herself to be competent in the use of the large sextants and quadrants, when Johannes published his Machina coelestis in 1673 he included two engravings showing Elisabeth using the octant and sextant with him respectively. These are the first printed images of a woman astronomer at work. They employed advanced astronomical instruments such as brass quadrants, sextants and octants (named for the amount of a circle they encompassed) and their observatory was a hub of innovation. The large sextants and octants required two people to operate. Each were equipped with an alidade, a sort of ruler to fix on the distant object being observed, and then its position could be read off. Johannes had invented his own precise alidade with cylinder to fix an objects position. With her knowledge of language, Elisabeth corresponded and struck up friendships with other astronomers. The French physicist and astronomer François Arago recounted that that she was both making useful observations and preforming calculations. In 1677 they were visited by King Jan III Sobieski, who gave them a stipend. Despite owning telescopes with which he had for instance, carefully observed the surface of the moon, Johannes was a hold-out where it came to making observations of stellar positions with telescopes, first employed for astronomy by Galileo Galilei (1564-1642) in 1609.  Johannes feared that telescopic observations might introduce distortions in locations. He is considered the last great astronomer to make observations with the unaided eye. 

This unusual attitude towards telescopes lead Johannes into controversy. In England, at the Royal Society, Robert Hooke accused him of making inaccurate observations and sent fellow astronomer Edmond Halley to visit Danzig, from May 26, to July 18, 1679. While Hooke was right about the utility of using telescopes, Halley was very impressed with the precision of Johannes and Elisabeth's naked-eye observations. Though privately he doubted naked-eye observations were the way to go, he confessed that in his controlled test, he could not determine which observations were more accurate. Six separate observers, including Johannes, Elisabeth and Halley with his with a 2-foot quadrant with telescopic sights made the same observations and Halley compared the data. Hooke was quite rude about the entire debacle, and Hevelius responded in kind. Other astronomers, like John Wallis, defended Hevelius because they were annoyed with Hooke. None of the members of the Royal Society involved in the bitter dispute came away with their reputations unbesmirched, with the exception of Halley, who had more diplomatic sense than the rest of them. That summer, they published Machina coelestis par posterior 'Astronomical instruments, second part' including a biography of Johannes, description of their instruments and 1,564-star catalogue of stars based on their observations.  

Then September 26, 1679 they suffered a devastating fire which destroyed their home, their observatory, their instruments, their library and printing press. Luckily, as Johannes wrote, before the fire he was feeling uneasy and "To lift my spirits, I persuaded my young wife, the faithful assistant for my nightly observations, to spend the night in our country retreat outside the walls of the city..." so they were unharmed. People present, including their 13 year old daughter Catherine Elisabeth managed to save several books including Kepler's works (purchased from his son), his new and improved celestial globe, thirteen volumes of his correspondence with scientists and royalty and the most importantly, the star catalogue by breaking into the burning house and throwing them out the windows! Luckily, dozens of copies of Machina coelestis which had already been sent out were not destroyed, but they lost everything else. Elisabeth had asked Halley to buy her a silk dress in London, in exchange for three of Johannes' books, and he wrote how when choosing the fabric he was uncertain if she would be in mourning because, having heard about the fire, astronomers in England feared Johannes had perished. Both King Jan III Sobieski and King Louis XIV of France sent them thousands of thalers to help them rebuild, and Sobieski granted him a yearly stipend of 1,000 Danzig gulden for the remainder of his life,  but it was nowhere near the estimated value of what they lost, which was over thirty thousand thalers. Undetered, they rebuilt their observatory by August 1781 and resumed work on their star catalogue, incorporating 341 stars only visible in the southern hemisphere, reported to them by their friend Edmond Halley. Elisabeth did much of the mathematical calculations and editing of the text. 

Johannes and Elisabeth Hevelius observing the sky with a brass octant (1673), Engraving from Johannes Hevelius' Machine Colestis: Pars Prior fig. O, facing p. 254. 

In 1685, Johannes published Annus climacterius 'Climactic Year' documenting their most recent observations and retelling the tale of the 1679 fire. Swiss mathematician and astronomer, Johann III Bernouilli wrote that Elisabeth contracted smallpox and was badly marked by it. Though Johannes had never contracted the disease he nursed her and never left her sick-bed. They began working on what would be Johannes' final publication. He died on his 76th birthday in 1687. After his death, she secured funding, took over the completion and publication of the Prodomus Astronomicae, 'The Elements of Astronomy". In 1690 she published the Prodomus Astronomicae, documenting two decades of observations, in three parts: the Prodomus was a preface with unpublished observations, which Elisabeth completed as well as writing and signing the dedication to the king as "Elisabeth, widow of Hevelius";  Catalogus Stellar Fixarum 'Catalogue of the Fixed Stars' (dated 1687) was a star catalogue including the positions and relevant data for 1,888 stars; and Firmamentum Sobiescianum sive Uranographia 'Sobieksi's Heavens, or a Map of the Heavens' (dated 1687) was a 56 sheet atlas of constellations for both northern and southern hemispheres from the catalogue complete with seven new constellations he delineated which are still in use (Canes Venatici, Lecerta, Leo Minor, Lynx, Scutum, Sextans, and Vulpecula) plus three which are now obsolete (Cerebus, Mons Maenads, and Triangulum Minus). Hevelius named one of these constellations for the king (Scutum Sobiescianum, or 'Shield of Sobieski' shortened now to Scutum) and one for his precious sextant, as well as several animals. It represented a significant advancement in astronomical observations and knowledge, containing both more stars and more accurate positions than Johan Bayer's Uranometria (based on Tycho Brahe's measurements). Elisabeth had not only ushered the book through publication, she participated in observations and calculations in a meticulous and systematic way. She died three years later, December 22, 1693 at 46 and was buried in the same tomb as her late husband. Arago wrote, "A complimentary remark was always made about Madame Hevelius, who was the first woman, to my knowledge, who was not frighted to face the fatigue of making astronomical observations and calculations." Arago was unaware of how Sophia Brahe (1556 or 1559-1643) had aided her brother Tycho, or of their near-contemporary Maria Cunitz (1610-1664), who was rather isolated from the astronomical community, so we should say that Elisabeth was amongst the first but not the earliest woman bravely facing late night observing and astronomical calculations. But her correspondence and friendships with fellow astronomers, as well as her collaboration with her husband, appearance in his books and her publication of the Prodomus, means that she and her accomplishments were recognized and remembered. A minor planet discovered at the Palomar Observatory in 1960 was named 12625 Koopman and a crater on Venus is named Corpman, a variant on the spelling of her maiden name, were named in her honour.

References

Ashworth, William B. Scientist of the Day - Elisabeth Hevelius. Linda Hall Library, University of Missouri-Kansas City. December 22, 2017.

Elisabeth Hevelius, Wikipedia, accessed February, 2026.

Jardine, Lisa. The Curious Life of Robert Hooke - The Man Who Measured London. Harper Collins. New York. 2003.

Masters, Karen. The Astronomers' Library. Ivy Press. London. 2024. 

Lutz, R.C. Elisabeth Hevelius, EBSCO Knowledge Advantage, 2022.

O'Connor, J.J. and E.F. Robertson. Elisabetha Koopman (1647-1493) - Biography. MacTutor. School of Mathematics and Statistics, St Andrews University. December, 2008.

O'Connor, J.J. and E.F. Robertson. Johannes Hevelius (1611-1687) - Biography. MacTutor. School of Mathematics and Statistics, St Andrews University. December, 2008.

Popova, Maria. Ordering the Heavens: Hevelius's Revolutionary 17th-Century Star-Catalog and the First Moon Map. The Marginalian. 2014.

Rosalyn Sussman Yalow

Rosalyn Sussman Yalow, linocut print, 9.25" 12.5" by Ele Willoughby, 2026

For the 10th #PrinterSolstice2526 prompt 'fraction' I made the portrait of the woman who taught us how to measure the fraction, or concentration of many things, like antigens like hormones to drugs to within our blood, and to diagnose and monitor related diseases, medical physicist and Nobel laureate Rosalyn Sussman Yalow. 

Rosalyn Yalow (née Sussman, 1921-2011) was born to a Jewish family in the Bronx during the Great Depression and attended Walton High School. Her parents had not had the opportunity to attend high school, but valued education and encouraged their children, Rosalyn and Alexander, to read books from the local library. Like Esther Lederberg, she got the opportunity for higher education at the all-female Hunter College, tuition-free. Her mother hoped she would become a teacher but Rosalyn opted to study physics.  She read Eve Curie's biography of her mother, double-Nobel laureate Marie Skłodowska-Curie, and Rosalyn got excited about nuclear physics. A lecture by Enrico Fermi in 1939, on nuclear fission, newly discovered by Lise Meitner, Otto Hahn and Fritz Strassman, captured her imagination. So, she became Hunter College's first physics major and even graduated early with honours. Money was tight, and opportunities for graduate school for a young Jewish woman were few, so she got herself a secretary to a biochemist at Columbia University, Dr. Rudolf Schoenheimer (who developed a technique of isotope labelling of biomolecules). She thought eventually she might go to graduate school at Columbia. She graduated Hunter College in 1941 She did not think that as a woman, she could get financial support for graduate school as she found a second secretary job. Columbia immunologist Dr. Michael Heidelberger hired her on condition that she study stenography. When the US joined WWII, suddenly there was more opportunity for women. Many male physics grad students were drafted or being sent to do secret wartime research and universities decided was better to admit women students, as men were otherwise occupied, than to have to close their doors. She was offered a teaching assistant position in the college of engineering at the University of Illinois Urbana-Champaign, where she was the first woman since 1917 and the only woman among 400 members. There were no women's washrooms, a real inconvenience when working late in the lab. She took extra undergraduate courses in physics to fill in gaps in her education. She was ambitious, and assertive and felt disapproval, particularly from other women, for not taking up teaching, the socially acceptable role for a woman at the time. She married fellow physics student, Aaron Yalow, the son of a rabbi in 1943. She learned how to build her own equipment and measure and work with radioactive substances.  She earned her PhD in 1945, then took further tuition-free physics back home in New York, under government auspices at New York University. 

Aaron knew he was not marrying a housewife, and supported her pursuing her career in physics, but he did not, on the other hand, help around the house. She valued the traditional roles of wife and mother. Her parents had not kept a kosher home, but she did for Aaron and later, their two children, Benjamin and Alanna. If Rosalyn had to travel, she would prepare his meals in advance, for him to warm up. Rather than the more modern ideas of work-life balance, Rosalyn's solution was simply to stint on sleep. Like many in her generation she was a bit wary of feminism and feminist organizations, placing a high priority on the role of wife and mother. She worried feminism lead women to eschew traditional duties. Conversely she advocated for more women in science and served as a loyal and dedicated role-model and mentor to younger women. She wrote, "The world cannot afford the loss of the talents of half its people if we are to solve the many problems which beset us." But, she did not want women in science to be treated differently from men so she disapproved of organizations for women in science. She recognized that WWII had created opportunities for women, but suggested that if the numbers of women decreased post-war, perhaps they were simply not interested. Nonetheless, she devoted her retirement years to advocating for better-quality child care and science education, aware that these could both make a significant positive impact on women's participation in science.

After grad school, Rosalyn got a job as an assistant electrical engineer at the Federal Telecomminations Laboratory, where she was the only woman. In 1946 she returned to Hunter College, where she taught physics. In 1947, Aaron introduced her to Columbia University medical physics Edith Quimby, who used radioactive isotopes. She asked to volunteer in Quimby's lab, anxious to get back into research; Quimby instead introduced her to influential medical physicist Gioacchino Failla. Failla immediately called the VA Hospital in the Bronx and got her a part-time job as a researcher there, to set up their Radioisotope Services. She set up a lab in an old janitor's closet, often building her own equipment, as the field was so new. She split her time between working in the lab and as a physics lecturer until 1950, because she wanted to keep mentoring students. One especially promising student had caught her attention: Mildred Spiewak (later the "Queen of Carbon Science" Dresselhaus), whom she persuaded to apply to graduate school for physics, rather than teacher's college. Rosalyn remained an active mentor to Millie through her career. 

Rosalyn Yalow with her research partner Solomon Berson, when she won the 1961 Ely Lilly Award of the American Diabetes Association. (via here)

The Veterans Association wanted to institute a research program on the medical uses of radioactivity, and by 1950, Yalow had equipped a radioisotope lab at the Bronx VA Hospital and she left teaching for research. There she met her longtime collaborator, physician Solomon Berson. He had already accepted a job with the VA Hospital in Bedford, Massachusetts, but after interviewing Rosalyn, they hit it off famously, and he changed his plans. Each happily married to someone else, they became "work spouses," an inseparable, complementary pair, in the lab for 22 years, using radioactive isotopes to study the human body. Sol taught Rosalyn immunology and Rosalyn taught Sol about radioactivity.

She wrote, "The only difference between men and women in science is that women have the babies. This makes it more difficult for women in science but, ... it is merely another challenge to overcome." The VA policy was that once women employees reached their fifth month of pregnancy, they had to leave, so, when expecting her children, she ignored the policy and deliberately neglected to inform the VA.

Rosalyn and Sol were interested in what happens to the insulin diabetics injected to allow them metabolize sugars. Rosalyn had a personal reason for her interest in addition to their scientific reasons; Aaron was diabetic. So they labelled insulin by adding a radioactive isotope of iodine, so they could follow the insulin in patients' bodies by the location of the radiation. Was the insulin being broken down by the body and passed through urine? To their surprise, diabetics who had been injecting insulin for some time, were retaining the insulin longer than health patients (themselves included amongst their healthy subjects). The insulin was being produced by cattle, and it was quite similar, but not identical to human insulin. They realized that the diabetics' bodies were detecting a foreign substance and producing an immune response to the cattle insulin, which was why it was processed more slowly and remaining in their bodies, but it also meant it was becoming less effective. These patients needed human insulin, and thus we now produce insulin which is identical to that naturally produced in healthy humans. 

They realized they could flip their method around to measure the insulin in blood. If they mixed a patient's blood with a mixture of radioactively tagged insulin and natural antibodies the antibodies would then pair with the tagged or patient insulin. Since the antibodies prefer the nonradioactive version of the insulin (or any other hormone they wanted to measure in this way), gradually, they separate from their radioactive partners and pair with the natural hormone. Then, they could separate all the hormone-antibody pairs (whether radioactive or not) and measure the radioactivity of the mixture. The difference between the radioactivity of the mixture from the original sample of tagged samples is a measure, or assay, of the amount of hormone in the blood sample naturally. Suddenly, rather than 100 cubic centimetres of blood, a diabetic patient's insulin level could be measured with a tiny blood sample. They proved that type 2 diabetes was caused by the body's inefficient use of insulin, rather than lack of insulin. Better yet, this method could be used with virtually anything which produced an immune response and did not require the patient to be exposed to any radioactivity. Yalow and Berson had developed the radioimmunoassay by 1960. Their methodology can be applied to measure minute quantities of hundred of substance, like hormones, vitamins, viruses, drugs and enzymes, and diagnostic hormone-related conditions and diseases, revolutionizing diabetes treatment. Their published many papers and trained many young scientists in their tight-knit "Mom and Pop" lab, by then grown much larger than the old janitor's closet.

Recognizing both its huge commercial potential and potential for improving or saving lives, Yalow and Berson refused to patent the method. Yalow was appointed a research professor in the department of medicine at Mount Sinai Hospital. Yalow won a Fullbright fellowship to go to Portugal, and the 1961 Ely Lilly Award of the American Diabetes Association.  In 1921, she won the Gairdner Foundation International Award for biomedical scientists advancing humanity and the American College of Physicians award. In 1972, she won the annual William S. Middleton Award for Excellence in Research, the highest honour of the Biomedical Laboratory Research and Development Service for outstanding biomedical science pertaining to the healthcare of veterans. She also won the Koch Award of the Endocrine Society. Tragically, also that year, Sol died at 54. Yalow named her lab for him and became the Solomon Berson Distinguished Professor at Large and became known at the "Mother of Endocrinology." Rosalyn knew that the Nobel Committee did not grant posthumous awards and would not generally award surviving researchers of joint research if one of the researchers had died. She knew she would have to prove herself on her own, and not simply as the "Mom" or their "Mom and Pop Lab." 

In 1975, Yalow and Berson (posthumously) were awarded the American Medical Association Scientific Achievement Award. In 1976 she became the first woman or nuclear physicist to receive the Albert Lasker Award for Basic Medical Research. In 1977, Rosalyn Yalow received the Nobel Prize in Physiology or Medicine for her role in developing the radioimmunoassay technique (along with Roger Guillemin and Andrew V. Schally, for their work on hormone production in the brain). She won the 1977 Golden Plate Award of the American Academy of Achievement. In 1978 she was elected Fellow of the American Academy of Arts and Sciences. She won the A. Cressy Morrison Award of the New York Academy of Sciences in 1986 and the National Medal of Science in 1988. She continued conducting research at her VA lab until she retired in 1991. She was inducted into the National Women's Hall of Fame in 1993. She died in the Bronx in 2011 at 89.

 

References

Berson, Solomon A., Rosalyn S. Yalow, Arthur Bauman, Marcus A. Rothschild, and Katharina Newly, Insulin-I'13 Metabolism in Human Subjects: Demonstration of Insulin Binding Globulin in the Circulation of Insulin Treated Subjects, The Journal of Clinical Investigation, February 1, 1956. 

Harrod, Jordan. Meet Nobel Prize winner Rosalyn Sussman Yalow, who let doctors see into your blood. Massive Science. May 12, 2019.

Rosalyn Sussman Yalow, Wikipedia, accessed February, 2026

Gellene, Denise. Rosalyn S. Yalow, Nobel Medical Physicist, Dies at 89. New York Times. June 1, 2011

Rosalyn Yalow. NobelPrize.org. Nobel Prize Outreach 2026. accessed February 2026.

Rosalyn Yalow and Solomon Berson, Science History Institute Museum & Library, accessed February, 2026.

The Nobel Prize in Physiology or Medicine 1977. NobelPrize.org. Nobel Prize Outreach 2026. accessed February 2026.

https://department.va.gov/history/featured-stories/dr-rosalyn-yalow/

https://www.acs.org/education/whatischemistry/women-scientists/rosalyn-sussman-yalow.html

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

Microbiologist Esther Lederberg, Printmaking to Replica Plating, the Lambda Phage and Bacterial Fertility Factor F

Esther Lederberg, linocut print, 11" x 14" by Ele Willoughby, 2026

The 8th #PrinterSolstice2526 prompt is two, so I thought of duplicates. This reminded me of trailblazing microbiologist Esther Miriam Zimmer Lederberg (1922-2006) whose pioneering contributions to bacterial genetics include such fundamentals as the first successful implementation of replica plating (with her first husband, Nobel laureate Joshua Lederberg) which allows microbiologists to reproduce a pattern of microbes on different Petri dishes and, for instance, compare their response to various things put in each plate. Reading about how they would press a velveteen-covered disk onto the primary plate with microorganisms and then imprinting the secondary plate (or plates) with the same distribution of microbes, I recognized this as printmaking, so I was delighted to read that her biographer believes she was inspired by having worked in her father's print shop as an adolescent. She also discovered the lambda phage and the bacterial fertility factor F. While she and Joshua both were awarded the Pasteur Medal from the Society of Illinois Bacteriologists, Esther's contributions, including work she did on her own, were often misattributed to Joshua, and she was not included in his Nobel win, nor was she even ever offered a tenured position. In life, she cared more about the science than her reputation, but today microbiologists are working to make sure she gets more of the recognition she always deserved.

Microbiologist Esther Lederberg (née Zimmer, 1922-2007) made discoveries fundamental to modern understanding of bacterial gene regulation, recombination and exchange, but her work was both overshadowed by, and sometimes misattributed to her male collaborators. 

Esther and her younger brother Benjamin were children of Orthodox Jewish Romanian immigrant, David and Pauline Geller Zimmer, herself the daughter of Romanian immigrants, who ran a print shop in the Bronx. Money was tight. The Zimmer children grew up helping in the shop during the Great Depression. David's siblings worked in the garment industry.  Esther had a talent for language. Esther pleased her maternal grandfather by learning Hebrew, unlike her cousins, and won awards for French and Spanish at school. She graduated from Evander Childs High School when she was only 15, and won a scholarship to attend Hunter College, where she planned to study French or literature. Once there, she chose to switch to biochemistry, though teachers warned she would have trouble finding a job, as a woman. Nonetheless she found work as a research assistant working on Neurospora crassa, a red bread mold,  with plant pathologist Bernard Oglivie Dodge. By the time she was 19 in 1942 she had graduated cum laude with a BSc in genetics. 

She got a position with Alexander Hollander at the Carnegie Institution of Washington (later the Cold Spring Harbor Laboratory), continuing her work on N. crassa and making her first scientific publication. She won a fellowship for Standford in 1944 to work with George Wells Beadle and Edward Tatum (who together would later go on to win the 1958 Nobel in Physiology or Medicine with her future husband Joshua Lederberg). Tatum told her would only teach her genetics if she could work out why one of the Drosophila fruit flies in a bottle had different coloured eyes; she did this so well, he made her his Teaching Assistant. After summer school at Stanford's Hopkins Marine Station with microbiologist Cornelius Van Niel, she entered the master's program, and completed her M.A. in genetics with her thesis "Mutant Strains of Neurospora Deficient in Para-Aminobenzoic Acid" in 1946. She married one of Tatum's students at Yale, Joshua Lederberg, and moved to Yale's Botanical Laboratory. Joshua completed his doctorate in 1947 and was offered an assistant professorship at the University of Wisconsin, so the couple moved to Madison, Wisconsin. Esther enrolled in the doctoral program and was awarded a fellowship by the National Cancer Institute. Joshua developed a reputation as a brilliant thinker while Esther developed a reputation as a skilled expert experimentalist and they worked well as a team. Her 1950 doctoral thesis under supervision of plant geneticist R.A. Brink was titled "Genetic control of mutability in the bacterium Escherichia coli". She had already made some foundational discoveries of the fertility factor, but Joshua insisted she focus on completing her PhD, rather than follow up with further experimentation. 

She spent most of the 50s at the University of Wisconsin, and in 1951 she discovered the lambda bacteriophage, a virus which infects and replicates within bacteria, and she published a detailed description in Genetics in 1953 with Joshua as the second author. He had asked her to wait before publishing because he felt it was not the top priority for the lab at that time. When Joshua won the 1953 Eli Lilly Award he told a reporter that Esther should have be recognized too. The lambda phage became a foundational discovery in molecular genetics, invaluable for understanding gene regulation and recombination. She incubated a mix of a parent E. coli K12 strain and a mutant E. coli K12 strain made with UV light and saw plaques, which indicated the presence of a bacteriophage. This type of virus would have been killed by UV light, so it must have come from the parent strain. She called this the lambda phage and found it had two lifestyles: either the typical, well-known phage lifestyle, the lytic cycle, where it rapidly made copies of itself inside the E.coli before bursting out (called lysis) or alternatively existing quietly as just another genetic marker within E. coli, without killing the cell.  With her husband, she found that the quiescent form genetically mapped near the genes in the E. coli which metabolize lactose sugar (gal). The Lederbergs proposed that the lambda's genetic material integrates into the E. coli chromosome next to the gal genes, which let it replicate as a prophage along with the host bacteria's own DNA (a process called lysogeny). This was the first time this 2-part cycle had ever  been described and today, the lambda phage is a key model for understanding other viruses (such as herpes simplex virus) which exhibit these split lytic-lysogenic lifecycles. To leave the host, the prophage needs to excise itself from the bacterial DNA and sometimes when this occurs the phage DNA is accompanied by some adjacent host DNA which the phage can then introduce into a new host; this is called specialized transduction.  Esther presented her lambda studies, including  λ lysogeny and specialized transduction in Canberra, at the Symposium of Bacterial and Viral Genetics in 1957, and her results on where it integrates, the fine-structure mapping of the gal locus in Montreal in 1958 at the 10th International Congress of Genetics.

While studying where the lambda prophage, she crossed bacteria with the prophage with strains without the prophage but will known genetic markers. She found that some crosses failed to form recombinants. She suspected something, a fertility factor, must be missing, writing that during her studies, "one day, ZERO recombinants were recovered... I explored the notion that there was some sort of 'fertility factor' which if absent, resulted in no recombinants. For short, I names this F." She published with Luigi Cavalli-Sforza and Joshua. She was right and others' later research showed that F was a DNA sequence with genes which allow a bacterium to donate DNA to a recipient bacterium by direct contact or conjugation. This DNA sequence can be either an independent plasmid or integrated into the cell's chromosome. Likely because of the delay between her discovery of the fertility factor F and her follow-up experiments and publications, this work is often unfairly primarily attributed to Joshua in textbooks. 

Microbiologists had been struggling to make identical geometrical configurations of microbes on multiple plates and tried using everything from toothpicks, paper, wire brushes and multi-pronged inoculators, until Esther and Joshua successfully implemented their simple solution to replica plating. Their process, pressing a plate of bacterial colonies onto sterile velvet and stamping them onto secondary plates of media with different ingredients to suit the traits needed for study (for instance, containing various antibiotics), mimics how a press works, like in Esther's father's shop! The first time she tried out the idea, she was inspired by her own powder puff, the velveteen pad in her makeup compact. It was, after all, designed to transfer powdered makeup to a person's face; maybe it could transfer configurations of microbes from plate to plate, so she tried it. Esther, whose family were garment workers, was the one familiar with textiles and she used this knowledge to source the best fabric for the job. Nonetheless, this method has also often been inaccurately credited to Joshua only. Since they could reliably replicate their plates, they were able to show, as previously demonstrated by Luria and Delbrück, that bacteriophage- and and antibiotic-resistant mutants spontaneously arose even in the absence of phages or antibiotics. These mutations were not occurring in response to their environment, as held by Lamarckism. Luria and Delbrück's work relied on mathematical arguments and had not gained traction but the simplicity of the Lederbergs' methods finally convinced their colleagues. Their method for replica plating remains a staple of microbiological laboratory work today.

In 1956, both Lederbergs jointly won the Pasteur Award for their contributions to science.  In 1958, Joshua Lederberg shared the Nobel Prize for Physiology or Medicine for discovering bacterial conjugation, with Edward Tatum and George Beadle for their work with genetics. Joshua had discovered that bacteria do not only make identical copies of themselves; they sometimes reproduce by mixing genetic material, as occurs in sexual reproduction, to make something new. Esther, whose work, like discovering the fertility factor with allows this genetic mixing to occur, was essential to, or made in collaboration with Joshua's research, was not included in his win. Attending the formal ceremony in Stockholm, by his side, in her gown, only playing the role of scientist's spouse, must have been bittersweet. Esther became perceived as merely his research assistant. Joshua thanked her simply for her "companionship" in his acceptance speech. Castelli-Sforza wrote, "Dr. Esther Lederberg has enjoyed the privilege of working with a very famous husband. This has been at times also a setback, because inevitably she has not been credited with as much of the credit as she really deserved. I know that very few people, if any, have had the benefit of as valuable a co-worker as Joshua has had."

In 1959 Esther and Joshua returned to Stanford, where Joshua became the head of the Genetics department. Esther could not be employed in the Genetics department because of anti-nepotism rules. Esther and two other women petitioned the dean of the medical school over the lack of female faculty. Eventually she got an untenured faculty position as a research associate professor in the Department of Microbiology and Immunology, where Esther worked for the rest of her career. This meant that her job was not secure and her short-term appointment was renewed on a rolling basis, dependent on her ability to secure grants. After their divorce in 1968, Esther had to fight to retain her position. She founded and directed the Plasmid Reference Center (PRC) from 1976 to 1986, where she organized, maintained, named, and distributed plasmids of various types including those coding for antibiotic-resistances, heavy-metal-resistance, virulence, conjugation, colicins, transposons and insertion sequences. Her sequential numbering system continued until her retirement. In 1974 she was effectively demoted and forced to transition to an adjunct professor of medical microbiology. She retired from the Department of Microbiology and Immunology in 1985 but continued as a volunteer at the PRC.

While not working as a scientist, Esther was a devoted amateur musician focusing on early music, playing medieval, Renaissance and baroque music on original instruments including several different sizes of recorder. She founded the Mid-Peninsula Recorder Orchestra, which plays music dating back to the 13th century, in 1962.  A lover of the works of Charles Dickens and Jane Austen, she belonged to the Dickens Society of Palo Alto and the Jane Austen Society. Through her pursuit of early music, she met fellow enthusiast, engineer Matthew Simon newly arrived at Stanford in 1989, and they married in 1993. She died in 2006 of pneumonia and congestive heart failure at age 83. Esther always thought the science itself was more important than her reputation, but her second husband understood that her work was under-appreciated because she was a woman. Simon has worked tirelessly to make an extensive memorial website for Esther, so her story and role in the development of microbial genetics is not forgotten. The wealth of biographical information and documents there from her career has helped her biographers and microbiologists working to tell a more complete and accurate story of the development of their field.

References

Esther Lederberg, Wikipedia, accessed February, 2026.

Barron, Madeline. Esther Lederberg and the Rise of Microbial Genetics. American Society for Microbiology. October 4, 2023. 

Joshua Lederberg, Wikipedia, accessed February, 2026.

Lederberg, J and E.M. Lederberg. Replica plating and indirect selection of bacterial mutants. J Bacteriol. 1952 Mar;63(3):399-406. doi: 10.1128/jb.63.3.399-406.1952. PMID: 14927572; PMCID: PMC169282.

Simon, Matthew. Esther M. Zimmer Lederberg Memorial Website. Accessed February, 2026

Steinmetz, Katy. Esther Lederberg and Her Husband Were Both Trailblazing Scientists. Why Have More People Heard of Him? Time Magazine. April 11, 2019.

"Women in Microbiology -Esther Lederberg" Live session from ASM Microbe 2017 hosted by Hazel Barton with guest Mark O. Martin interviewed by Rebecca V. Ferrell.

Zeldovich, Linda, Esther Lederberg changed our understanding of how bacteria breed. Popular Science, May 23, 2022.

Agnes Pockels Find Wonder While Washing Dishes and Invents Surface Science

Agnes Pockels, linocut, 11" x 14" on Japanese paper, by Ele Willoughby, 2026

The seventh prompt for #PrinterSolstice2026 is division, and I have decided to focus on the dividing line or interface, and the self-taught scientist who pioneered surface science. 

When Agnes Pockels (1862-1935) was 18, she did not get the chance to go to university and study physics like her brother Friedrich, who became a professor of theoretical physics and discovered the Pockels effect studying optics and electromagnetism. She took her "passionate interest for natural science" to those household chores she was allowed to pursue, and applied her analytic mind and careful observation to what she saw even in greasy dishwater. She became fascinated with how soap behaves on the surface of the water, and especially in the effect of impurities. She began performing experiments as an amateur chemist. In so doing, she pioneered the entire field of surface science describing physical properties of liquids and solids at interfaces, gained international recognition, published a series of peer-reviewed papers and earned an honorary doctoral degree from Braunschweig University, Germany. Performing kitchen-table research, she developed a surface film balance technique to study soap and surfactants at the air-liquid interfaces and defined the Pockels point, the minimum area a single molecule can occupy in a monomolecular film.

Agnes Pockels' first paper 'Surface Tension'
published in Nature in 1891, complete
with Lord Rayleigh's introduction.

Born in Venice in 1862, when it was part of the Kingdom of Lombardy-Venetia, Agnes' father Theodor served in the Austrian army. Malaria was an ever-present risk in the region and struck the family. When Theodor fell ill in 1871, he retired and moved the family, his wife Alwine, daughter Agnes and son Friedrich, to Brunswick, in the newly formed German Empire. Agnes was interested in science, and attended the Municipal High School for Girls. Though she would have loved to pursue higher education in science, and especially in physics, women were not admitted to German universities (with a very few notable exceptions). So, she studied science at home, while caring for her parents for three decades, as expected for an unmarried daughter. When her younger brother Friedrich went to study physics at the University of Göttingen, he would share his textbooks with her. Later, he would supply her with research in the academic literature to help her pursue her own self-taught studies. When the universities finally began admitting women, Agnes followed her father's wishes and refrained from attending so she could continue keep house and to care for her sick parents.

She began experiments when she was 18, and by age 20 in 1882, she had devised a slide through for making quantitative measurements of soapy water and other materials. Her water-filled through was 70 cm long, 5 cm wide and 2 cm deep and she laid a metal strip (about 1.5 cm wide) on the water, across the width of the through, so she could divide the surface into two parts. A lengthwise ruler along the through allowed her to precisely measure the surface area of both sides as the strip was moved along the length of the through. Building on the plate method of Ludwig Wilhelmy, she devised a simple but clever means of measuring surface tension in either area with a ceramic button (6 mm in diameter) placed on the surface. She would raise the button with an apothecary's balance or weighing scale to measure the force required to lift it from the water. She could then compare the force required to lift her button from pure water, or water with various added substances and impurities. Using her through, she investigated surface forces of monomolecular films, emulsions, solutions and the effect of impurities on physical properties and she came to understand surfactancy, the property of a chemical compound to decrease surface tension at an interface between different materials (at least one of which is a fluid).  She found that small amounts of impurities could have a large impact on surface tension. 

Agnes Pockels in 1885

Her through influenced future experimentalists in colloidal and surface science who employ the modern Langmuir-Blodgett through, an improved apparatus based on her work, still in use today. When Irving Langmuir won the 1932 Nobel Prize in Chemistry for his work in the surface chemistry of oil films, he was building on the 18 year old autodidact Agnes Pockels' experiments with a button and thin tray, performed in her kitchen (not that he bothered to mention this in his Nobel lecture). Her sister-in law, Elisabeth Pockels wrote, "every day, millions of housewives were unhappy to see greasy dish water and just wanted to get rid of it, but it inspired this very person to make observations and finally also work on scientific treatises." She made her meticulous investigations for a full decade without any communication or collaboration with the world of academic science. She had tried writing to physicists at the University of Göttingen about her work but received no reply. Then in 1890, her brother sent her a paper on surface phenomena by renown physicist Lord Rayleigh.  With her brother's encouragement, she wrote Rayleigh a modest, 12 page letter, including all her results. She explained that she had been unable to publish her results in any scientific journal and she gave him her permission to use her results if they were at all useful. Rayleigh did not speak German, but luckily, his wife Evelyn could translate the letter for him. He instantly recognized the importance of Agnes' work and to his credit he sent them to Nature, putting the weight of his reputation behind supporting the work of an unknown amateur, writing, "I shall be obliged if you can find space for the accompanying translation of an interesting letter which I have received from a German lady, who with very homely appliances has arrived at valuable results regarding the behaviour of contaminated water surfaces. The earlier part of Miss Pickles' letter covers nearly the same ground as some of my own recent work, and the main harmonises with it. The later sections seem to me very suggestive, raising if they do not fully answer, many important questions. I hope soon to find opportunity for repeating some of Miss Pockels' experiments." Nature published Ages' letter with Rayleigh's cover letter. This paper, 'Surface Tension' is considered a landmark in the history of surface chemistry.

Seeing her research published encouraged her to continue her experiments, and her correspondence with Rayleigh. She wrote to him about the paramount importance of cleanliness in these experiments and her recognition of how contamination had hampered her previous efforts. Even dust could hinder reproducibility. To study monolayer films, she developed a procedure for the deposition of the compound of interest as a solution in benzene on the surface of the water in her trough and she was able to measure layer thickness at 13Å (an Angstrom is a billionth of a metre, roughly the size of a single atom). She published her second paper, with Rayleigh's help, in Nature in 1892. She went on to study forces on monomolecular films, the calming effect various oils can have on water, various liquids and their adhesion on glass and the surface tension of emulsions and solutions, other surface phenomena like capillary and contact angles, publishing her results in scientific journals including Nature, Wissenschaftliche Rundschau, Annalen der Physik, and Science.  She was invited to give public lectures by German universities.  She published subjects beyond surface science, including her translation of Georg Howard Darwin's 'The Tides and Kindred Phenomena in the Solar System' and a philosophy paper in Annalen der Naturaphilosophie. 

German scientists finally took notice. At the Physikalische Institut of Techniche Hochschile Brauschweig (the Physical Institute of the Technical University Brauschweig), crystal physicist Woldemar Voigt (1850-1919) offered her lab facilities, so over the next decade, she squeezed in further research there around her responsibilities at home. 

Soaps are surfactants made of molecules which have a "water loving" hydrophilic head and a water-repelling hydrophobic tail which is insoluble in water and tends to sit on its surface. Using her sliding through, Agnes found the tiny addition of surfactants had a small impact on surface tension until the slider passed a certain point, where tension would suddenly increase. She plotted surface tension versus slider position, showing the compression isotherm. She realized that the point the compression isotherm abruptly changed was when the continuous film of surfactant was a single molecule thick. With further experimentation she found that the threshold point is the same for a variety of surfactants. Knowing the volume of surfactant soap and the surface area covered by the film, she was able to calculate the surface area of the water occupied by a single molecule to be 20 square Angstroms, now known as the Pockels Point.

Early in the new century, in 1902, her parents' worsening health interrupted her work at the university and she became their full-time carer. Her father died in 1906, her brother died in 1913 and her mother died in 1914. She herself was in quite poor health and she needed to have a stay in a sanitarium. Then during the first world war, she lost contact with the scientific community. Germany was isolated and she was unable to access the foreign scientific literature. Her health and eyesight deteriorated. She had invested wisely and was more financially insulated than other Germans from the post-war financial chaos. She felt the responsibility to protect those around her from the threat of hunger and homelessness and she stopped carrying out her experiments. She published the last of her 14 papers, reviewing her classic work, in 1926. She spent her later years travelling in Europe and as the devoted Auntie Agnes to her brother's children. She finally received public recognition for her work, winning the Laura R. Leonard Prize of the Colloid Society in 1931. In 1932, Techniche Hochschile Brauschweig granted her an honorary doctorate in engineering. She was the first woman to receive such an award. Legendary German colloid chemist Wolfgang Ostwald published her biography on her 70th birthday in 1832. Her sister-in-law Elisabeth Pockels also wrote a biography of Agnes, focusing on her personal life. Since 1993 the university has granted the Agnes Pockels Medal to people who have made outstanding contributions to the development of the university, promoting research and teaching, particularly by women. In 2002 Techniche Hochschile Brauschweig established the Agnes Pockels Laboratory to foster chemical education and aid chemistry teachers, focusing on children under 10, especially girls.

Despite her limited access to education Agnes Pockels found wonder in a mundane tub of dishwater, and launched an entirely new field of science.

References

Agnes Pockels, Wikipedia, accessed January, 2026

DeBakcsy, Dale. How a Kitchen Experiment Spawned a New Science: The Surface Physics of Agnes Pockels. The Women in Science Archive. October 16, 2023.

Derrick, M. Elizabeth. Agnes Pockels, 1862-1935, Journal of Chemical Education, vol. 59, no, 12, pp. 1030-1031, December 1982

Gratzer, Walter (ed.). A Beside Nature - Genius and Eccentricity in Science 1869-1953. W.H. Freeman and Company, New York. pp. 88. 1999.

Kruse, Andrea and Sonja M. Schwarzl. "Zum Beispiel Agnes Pockels," Nachrichten aus der Chemie, 06, 2002, translated as "Who was Agnes Pockels - Agnes Pockels - Housewife and Chemist", Braunschweig Technical University, accessed January, 2026.

RAYLEIGH Surface Tension. Nature 43, 437–439 (1891). https://doi.org/10.1038/043437c0

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.