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

Tiny prints and Unity and Proportion

There's a nifty thing (and hashtag) on Instagram, started by printmaker Kat Flint (@flintkat): #tinyPrintTuesday. There's all these delightful tiny prints in my feed. I wanted to get in on it, and have decided to make tiny linocut prints of animals I have seen in my tiny backyard. I haven't set a specifically size, yet, but these are no more than 4 inches across. It's a great way to use tiny leftover bits of lino and scraps of paper. So far, I have made a cottontail rabbit, skunk and opossum.

Skunk, linocut by Ele Willoughby, 2023

Opossum

 linocut by Ele Willoughby, 2023

Cottontail, linocut by Ele Willoughby, 2023

I have also made a couple of prints for the #printerSolstice series of prompts. For Proportion, I combined my existing golden spiral lino block with my own hand print. For Unity, I had fun once again playing with the concept both literally (in mathematics, unity means 1) and figuratively (with unity of design). I built up a background of radiating colours with my gelliplate and added a linocut 1.

Proportion, 9" x 12" linocut and handprint by Ele Willoughby, 2023

Unity, 8" x 10", linocut and gel plate print by Ele Willoughby

Kathleen Lonsdale, crystallographer, pacifist and prison reformer

 

Kathleen Lonsdale, 11" x 14" linocut print on Japanese kozo by Ele Willoughby, 2023

I chose Kathleen Lonsdale DBE FRS (née Yardley, 1903-1971) for the #printerSolstice prompt shape because she solved a longstanding chemistry conundrum of the shape of benzene & her drawing of electron density for hexachlorobenzene (green) & model of hexamethylbenzene explore shape in different forms. 

Going to Holloway prison was the single most formative experience for Kathleen Lonsdale’s scientific career and it gave her the ability to speak to anyone. Her husband said, “Before prison it might have bothered her to go to Buckingham Palace. Afterwards, Holloway or Buckingham Palace were all the same.” Born the tenth child of a poor family in Ireland, with four brothers who died in infancy, pacifist, prison reformer and physicist Dame Kathleen Lonsdale DBE FRS (née Yardley, 1903-1971) served time, as she was unwilling to compromise her beliefs. She was also a trail blazing crystallographer who solved a conundrum which had plagued chemists for decades: the shape of the benzene ring, proving it was flat using x-ray diffraction on hexamethylbenzene in 1929. She was the first to employ Fourier spectral methods and used them to solve the structure of hexachlorobenzene in 1931. In 1945 she was one of the first two women elected Fellow of the Royal Society and was the first woman in several roles including: tenured professor at University College London, president of the International Union of Crystallography and president of the British Association for the Advancement of Science.

Her father, a soldier and then a postmaster Harry Yardley and her mother, a strict fundamentalist Baptist of Scottish descent Jessie Cameron did not have a happy marriage. Between the unrest in Ireland and her father’s alcoholism, her mother decided to divorce and move the children to Seven Kinds, Essex, England when Kathleen was five. She won a scholarship to the Ilford County High School for Girls but had to go to the boys’ school in her final two years as the girls’ school did not offer mathematics and science. During WWI her home was on the Zeppelin route; she did homework by candlelight during air raids and first developed her opposition to war. Anxious to get to university as soon as possible she went to the Bedford College for Women in London on a county scholarship. After her first year she won a university scholarship and switched from mathematics to physics, against all advice (especially that of her old headmistress who told her she would never distinguish herself in physics). She graduated in physics with the highest score ever for a London University, with a BSc in 1922, which brought her to the attention of physics Nobel laureate William Henry Bragg, one of her examiners.

Bragg offered her a spot on his team at University College (and then the Royal Institution), and a grant of a £180 a year! She lived at home and contributed to family expenses, gaining her MSc from University College London in 1924. She worked with Bragg until she married in 1927 and followed her husband, research chemist Thomas Lonsdale to Leeds, where he had been offered a job at the Silk Research Association. Shortly after her marriage, she applied for an 1851 Exhibition Fellowship which Bragg expected she would win, as several of his other (all male) students had done. Not only did they turn her down for the award, they wrote they “would be breaking the spirit of the regulations in awarding an exhibition to a married woman.” Luckily Bedford College offered her a research grant and she continued to correspond with Bragg. She worked part-time as a physics demonstrator and doing lab work in Leeds. It was here that she was given crystals of hexamethylbenzene, the first important structure she solved. Debate had been raging between organic chemists and crystallographers whether benzene was flat or  had a zigzag shape like cyclohexane, but benzene itself was a challenge to crystallize. Lonsdale had the insight that she could instead look at the benzene within hexamethylbenzene and in the process of solving its form, she proved that the benzene ring (which it contained) was flat. She followed this with solving the structure of hexachlorbenzene; this was important as she was the first to investigate an organic compound with Fourier analysis. She had cleverly found a project she could do with calculations rather than lab work while she focused on starting their family. She also developed popular crystallographic reference tables with W.T. Astbury. She considered giving up science, but Thomas supported her research told her he “had not married to get a free housekeeper.” He encouraged her to continue in research. When they had their first daughter in 1929, Bragg convinced the Royal Institution to grant her £50 to employ some childcare  so she could work on calculations. Then they moved back to London for Thomas’ new job, and had a second daughter in 1931. Bragg, anxious to have her back, was able to find a further £200 to assist her at home so she could  return to work in 1931. They had their son in 1934. She earned her doctorate from the University of London in 1936 while working at the Royal Institution, where she stayed for 15 years. She worked with Bragg until his death in 1942, then with Sir Henry Dale, as a Dewar Fellow from 1944 through 1946.

She was raised a Baptist, but in 1935, she and her husband, both committed pacifists, became Quakers. She became a Sponsor of the UK Peace Pledge Union, which meant she signed the pledge "War is a crime against humanity. I renounce war, and am therefore determined not to support any kind of war. I am also determined to work for the removal of all causes of war.” They turned the top floor of their house into a flat where they welcomed refugees from Germany. When she was required to register for civil defence duties during WWII she refused to do so and refused to pay the small fine for not registering. She believed there should be an exemption for conscientious objection. She was sentenced to serve a month in Holloway prison, where the grim conditions lead to a life-long commitment to prison reform. While she was imprisoned, she found the clothing unclean, medical exam sketchy and she collapsed under her workload, scrubbing and cleaning; only then did they lighten her workload. Sir Henry Dale requested that she be given access to papers and instruments and she was allowed to work in her cell in the evenings.  Her colleagues worried she would be bored; she was in fact absorbed, talking to fellow prisoners about their lives and crimes. Her second fine for refusing to register for civil defence was paid anonymously, much to her chagrin; she would have rather stood on her principles and serve another prison term. She contributed to a pamphlet on Prison for Women about her experiences in Holloway and the need for prison reform.

In 1945 Lonsdale and Marjory Stephenson were the first women elected Fellows of the Royal Society. Then she finally got a permanent position. In 1946 she was appointed Reader in Crystallography and then Professor of Chemistry and Head of the Department of Crystallography at the University College London in 1949, finally beginning to teach and run her own research group, mentoring future crystallographers. She was their first tenured female professor. She researched the use of x-ray imaging at different temperatures and the structure and texture of crystals. She worked on the synthesis of diamonds. She won the Royal Society’s Davy Medal for significant discoveries in chemistry in 1957. Later she worked on solid state reactions, pharmacology and structure of methonium compounds and stones and minerals produced by the human body like kidney stones. She became an emeritus professor after 1968. Nobel laureate Dorothy Hodgkin wrote, "There is a sense in which she appeared to own the whole of crystallography in her time.

In 1953 she delivered the keynote Swathmore Lecture at the Yearly Meeting of British Quakers, “Removing the Causes of War”. She wrote about peaceful dialogue was appointed the first secretary of Churches' Council of Healing by the Archbishop of Canterbury. 

When Thomas retired at 60, they moved to Brexill-on-Sea; this meant 5 hours a day commute for Kathleen, but she felt it worthwhile though she was tired. Thomas helped with her tremendous amount of correspondence about peace and prison reform, and would bring her dinner in bed as soon as she got home. She was someone who never stopped working, even when she became ill and was hospitalized. She died in hospital 1971 from anaplastic cancer, at age 68, the day after Thomas' 70th birthday.

References

Kathleen Lonsdale, Wikipedia, accessed February, 2023

Hodgkin, Dorothy M.C., Kathleen Lonsdale, 28 January 1903 - 1 April 1971, Biographical Memoirs of Fellows of the Royal Society, Volume 21, Issue 21, November 1975

Melinda Baldwin, The Royal Society’s first woman physicist, Physics Today, 25 January, 2018. DOI: 10.1063/PT.6.4.20180125a

One crystal model of hexamethyl benzene, Science Museum Group, Object Number: 1993-421/4/11, Gift of University College London, in memory of Dame Kathleen Lonsdale 

Vera Rubin and What Can't Be Seen

Vera Rubin, linocut on Japanese paper, 11" x 14", by Ele Willoughby, 2023. The orbital velocity of galaxies is plotted against distance (on top of a galaxy, from its centre outwards). There's a wide gulf between the drop off that would be predicted by Kepler's laws and what was observed. Here, I have left that region white to emphasize that it was evidence of something missing and unseen. So I selected her for the #printerSolstice prompt space, so I could allude to outer space and have the space within the composition be what was telling the story.

This is a linocut print of renown astronomer Vera Rubin (neé Cooper, 1928-2016) and her discovery that the angular motion of galaxies deviates considerably from predictions, which we now know was the first evidence for dark matter, confirmed in the decades since. 

Her parents, Eastern European Jewish immigrants, electrical engineer Pesach Kobchefski (anglicized to Philip Cooper) from Lithuania, and Rose Applebaum from what is now Moldova, met in Philadelphia, working at Bell Telephone; though her mother's job ended when she married. When Vera was 10 they moved to Washington, DC, where she watched stars from her window and first fell in love with astronomy. “What fascinated me was that if I opened my eyes during the night, they had all rotated around the pole and I found that inconceivable. I just was captured,” she later told AIP in 1995.  With her father she built a simple cardboard telescope and tracked meteors. Her older sister went to law school. After she finished high school in '44 she ignored her physics teacher's advice to pursue art rather than science and went to the women's college Vassar, where astronomy trailblazer Maria Mitchell had been a professor as early as 1865. She graduated with honours, the sole astronomy graduate of 1948.

She wanted to pursue graduate studies at Princeton but was barred due to her sex; Princeton took 27 more years to admit women astronomy graduate students. She turned down an offer from Harvard, and instead opted for Cornell where her new husband, physicist Robert Joshua Rubin was a graduate student. During her masters (Cornell, 1951) she studied the motions of 109 galaxies. Hubble flow (or the Hubble-Lemaître law) states that galaxies are moving away from us at speeds proportional to their distance. Rubin was one of the first to observe a deviation from this law. She studied under Philip Morrison, Hans Bethe, and Richard Feynman and worked with astronomer Martha Stahr Carpenter to find a thesis topic on galaxy dynamics. She said that Carpenter's "course in galaxy dynamics really set me off on a direction that I followed almost my entire career.”  Her husband brought her a paper by renown physicist and cosmologist George Gamow who pondered whether galaxies moved like solar systems and it inspired her to start investigating how galaxies move. She found a plane of higher density of galaxies, which years later we would recognize as was some of the earliest evidence of the super galactic plane, the equator of our supercluster of galaxies. 

One of her advisors, Robert Shaw told her that her work was sloppy but should be presented to the American Astronomical Society (AAS) meeting. Since she was not a member, and very pregnant, he could do that - under his own name, not hers. So, she said she could go. She found the discussion after was acrimonious and she felt like an imposter. Her paper was never published. 

She took 6 months maternity leave but found it immensely difficult being at home with their lovely baby but watching her husband going to work daily to pursue what he loved. It was her husband who insisted she return to grad school. He took a job at the National Bureau of Standards in Washington, D.C. She gained experience working summers at the Naval Research Laboratory and the US Naval Observatory. She was admitted to the PhD program at Georgetown University, the only university in Washington, D.C. with a graduate astronomy program, at age 23, expecting their second child. The Jesuit astronomer Fr. Francis Heyden taught his courses at night, a real challenge with a young family. She encountered sexism, and recounted how she was not allowed to meet her advisor in his office as women were barred from that area of the Catholic university. When writing her thesis, Heyden got her in contact with George Gamow, who worked at the nearby Applied Physics Laboratory and was an adjunct professor at George Washington University. Gamow took her on as a student. In her 1954 thesis she noted that galaxies clump together rather than being randomly distributed - a largely ignored idea it took the field decades to catch up with. 

While her four children were very young, she taught at Georgetown and Montgomery College for several years before gaining a research position at the Carnegie Institution of Washington's Department of Terrestrial Magnetism (which operated the Wilson Observatory in California and had a new high-tech magnetically focused electronic image tube which could increase the sensitivity of telescopes). She worked for a year with Geoffrey and Margaret Burbidge observing rotating galaxies using the McDonald Observatory's 82" telescope. She was the first woman to use the Palomar Observatory in 1965, pragmatically solving the lack of washrooms by claiming one by going to her room, cutting out a little paper skirt and pasting it to the little man icon on the door.  "There you go; now you have a ladies' room." At Carnegie she met physicist and astronomical instrument maker Kent Ford. Together they made the most sensitive spectrometer of the day using the magnetically focused image tube- an instrument which divided light into its constituent frequencies and importantly allowed astronomers to study small regions of galaxies previously too dim to observe, not just galaxies in their entirety. They started looking at the newly discovered quasars but she did not enjoy the competition from astronomers with more access to world-class telescopes and the race to explain these objects. She wanted to carve out a niche to themselves. They decided to look at Andromeda Galaxy, returning to her interest in galaxy dynamics with Ford's spectrometer allowing them to see if galaxies did rotate like our solar system. Since mass and hence gravity is clustered in the centre, nearer objects should go faster than objects at the periphery. But, when they looked at areas of hydrogen gas where new stars form, at various distances from the centre of the galaxy, they all seemed to be going at the same speed. The expected drop off with distance simply wasn't there. They spent years on the project, travelling to various telescopes across the country for observing time. Rubin spent long hours analyzing data on punchcards and always seeing the same thing: no drop-off with distance from the centre of Andromeda. So they looked at other galaxies, and more and more galaxies. They gathered dozens of rotation curves and they were all flat. It contradicted theory and they did not know why but their data was undeniable. (You can see a video of how galaxies were predicted to move next to how they are observed to move here).

The concept of dark matter was proposed by Jan Oort (1932) and Fritz Zwicky (1933) to explain how physics seemed to imply more mass than astronomers could see, but they were largely ignored and no one has developed any theory of how galaxies who behave in the presence of dark matter, nor had anyone gathered observational evidence of dark matter. Rubin and Ford simply did not know what their observations meant. "One day I just decided that I had to understand what this complexity was that I was looking at, and I made sketches on a piece of paper, and suddenly I understood it all," Rubin said. A halo of dark matter - that is, matter which is not luminous, which we cannot see with telescopes, perhaps better imagined as invisible or unseeable rather than "dark," around galactic cores would spread out the mass throughout the galaxies, and hence and speeds would remain flat with distance from the centre. This unseen matter that Rubin and Ford first observed is now understood as the stuff that dictates how galaxies move, and even the origin and fate of our universe. 

Since their discovery, a theoretical frame work was set out which fits their model and the Planck satellite measured dark matter by observing the cosmic microwave background. It imaged clumping in the early universe which otherwise would have been homogeneous but which instead, because of this dark matter, evolved into the superclusters of galaxies we know today. We now believe there is five times as much invisible dark matter and the luminous matter we can see. The discovery of dark matter revolutionized astronomy and lead to entire new subfields of astronomy and particle physics. She was a favourite to win the Nobel Prize for many years, but died before that ever happened. Twenty years after Rubin's research revealed dark matter, dark energy was discovered, and its discoverers received the Nobel in 2011. In 2019, three years after her death, James Peebles shared the Nobel Prize in physics for work on evolution of our universe- notably theoretical work on existence of dark matter and dark energy. Many physicists and astronomers lamented the egregious snub of Vera Rubin, by waiting until she had died rather than including her.

She also found evidence that some stars and gas within galaxies move counter to the prevailing motion, some of the first evidence of galaxy mergers.

Throughout her career she was a champion of women in science, writing, “I live and work under three basic assumptions. One: There is no problem in science that can be solved by a man that cannot be solved by a woman. Two: Worldwide, half of all brains are in women. Three: We all need permission to do science, but, for reasons that are deeply ingrained in history, this permission is more often given to men than to women.” She likewise championed scientific literacy.

She published more than 100 peer reviewed scientific papers, a collection of essays, was on the editorial boards of journals and a member of the National Academy of Sciences (the second woman astronomer admitted, after Margaret Burbidge) and won the National Medal of Science. She won the gold medal of the Royal Astronomical Society in 1996; she was only the second woman to do so, 168 years after Caroline Herschel. Carnegie named a post-doc fellowship in her honour and the American Astronomical Society named a Vera Rubin Early Career Prize. There is Vera Rubin Ridge on Mars and Asteroid 5726 Rubin, a satellite and the Vera C. Rubin Observatory named in her honour. All four of her children grew up to be PhD mathematicians and scientists and they credit their mother for making it look like desirable and fun. 

References

Vera Rubin, Wikipedia, accessed January 2023.

Meet Vera Rubin, November 17, 2021, Air And Space Museum, Smithsonian Museum.

Sarah Scoles, How Vera Rubin confirmed dark matter, Astronomy, Tuesday, October 4, 2016 

Matt Schudel, Vera Rubin, astronomer who proved existence of dark matter, dies at 88, Washington Post, December 26, 2016

Rachel Feltman, In memory of Vera Rubin, the woman the Nobel Prize forgot, Popular Science, December 27, 2016

Ethan Siegal, Who Really Discovered Dark Matter, Fritz Zwicky or Vera Rubin? Forbes, August 24, 2021

Chanda Prescod-Weinstein, The Disordered Cosmos, Bold Type Books, New York, 2022.

Kelsey Johnson, We're Sorry, Vera Rubin, Scientific American, October 16, 2019

Shannon Connellan, Nobel Prize in Physics awarded to scientists, some rally behind one who never got one, Mashable, October 8, 2019

Jack Frost, astronomers Kepler and Lepaute and bees in snail shells

 I haven't kept up with posting all my recent prints, so today, we're playing catchup! I've been doing #printerSolstice so I have managed to make a print weekly tie to their prompts. This year the prompts are elements of art and design: value, form, line, balance, texture and upcoming are space, shape, contrast, proportion, unity, pattern and variety. I'm trying to interpret these prompts in light on my ongoing series of prints of various sorts. The first one, value (or the lightness and darkness of colours), I applied to another slightly sinister winter folktale: Jack Frost and made a print in tints and shades of cobalt blue.

Jack Frost, linocut on cardstock, 5" x 7" by Ele Willoughby, 2022

For form, I thought of Kepler and how he arrived at his laws from thinking about music and then the Platonic Solids!

Johannes Kepler, linocut, 11" x 14" on Japanese kozo paper, by Ele Willoughby 2023

This is my linocut of mathematician and astronomer Johannes Kepler (1571-1630). We remember him for his role in the Scientific Revolution, and his three laws of planetary motion in particular. His laws modified Copernicus’ heliocentric model; he replaced the circular orbits with elliptical ones & described velocities of planets. Today we know them as:

1) Planetary orbits are ellipses with the Sun as one of the foci (top magenta ellipse)

2) A line from Sun to planet sweeps out equal areas in equal time periods (middle ellipse)

3) The square of the planet’s orbital period (or year) is proportional to the cube of the semi-major axis (shown as the horizontal arrow in the bottom ellipse).

But, I find it fascinating- & important to note- that he came to these laws exploring mystical ideas about music, geometry and congruence with physical phenomena. Sometimes we tell simple, but misleading stories about scientific progress. 

First he argued that the spacing of the 6 known planets from the Sun were related to the 5 Platonic solids, each encased in a sphere and nested one inside another. He had to order them selectively: octahedron, icosahedron, dodecahedron, tetrahedron and cube. He then related the size of the spheres to the  orbital periods of the planets (Mercury, Venus, Earth, Mars, Jupiter and Saturn). But this formula was not precise enough…. But we can see this as the seed of his 3rd law. The gold shapes are the nested Platonic solids from his Mysterium Cosmographicum. 

He also took the medieval idea of the “music of the spheres” literally and translated planetary angular speed as measured from the Sun as musical notes and finds that the minimum and maximum speeds of neighbouring planets approximate harmonies. Though unrelated to our modern ideas about our solar system these explorations of geometry and music ultimately lead to his correct models, which in turn were a significant steps towards Newton’s Law of Universal Gravitation.

Next came line, and I made it about the line traced by an eclipse:

Nicole-Reine Lepaute, linocut, 11" x 14" on Japanese kozo paper, by Ele Willoughby, 2023

This is my linocut portrait of Nicole-Reine Lepaute, née Étable de la Brière, (5 January 1723 – 6 December 1788). She was a French astronomer, mathematician and human computer. My print celebrates how she calculated the path of the solar eclipse of 1764. She also worked with Alexis Clairaut and Jérôme Lalande to much more precisely calculate the date of the return of Halley’s Comet. This is no mean feat when you realize this was essentially solving the notorious three-body by hand (as the gravitational pull of Jupiter and Saturn affect its orbit around the sun). They worked in parallel, calculating for 6 months straight, barely stopping to eat! She also produced astronomical almanacs from 1759 to 1783 and was also a member of the Scientific Académie de Béziers.

Some of the historic women of science whose names and achievements were recorded, are known to us because of their wealth and privilege. Though Nicole-Reine Lepaute was born in Luxembourg Palace, she was not an aristocrat; she was the sixth of nine children of the valet of the duchess de Barry and her sister. A bright child, she was self-taught and devoured all the books in the library. Her later friend and long-time collaborator, astronomer Jérôme Lalande wrote that she had "too much spirit not to be curious." 

She married the royal clockmaker Jean-André Lepaute, in the Luxembourg Palace, in 1764. She became responsible for the household accounts but her marriage also allowed her to pursue her interest in mathematics and astronomy. She applied her skills to document, observe and calculate the workings of all her husband's inventions. The Académie des Sciences sent Lalande to inspect her husband's new type of pendulum clock. The three worked on the theory of clockmaking and added to her husband's "Traité d'horlogerie," which he had published in 1755. Though she was not included as a co-author, Lalande was nothing but praise for her, writing, "Madame Lepaute computed for this book a table of numbers of oscillations for pendulums of different lengths, or the lengths for each given number of vibrations, from that of 18 lignes, that does 18000 vibrations per hour, up to that of 3000 leagues"

Though Newton's Law of Universal Gravitation, published in 1687, allowed astronomers to calculate planetary orbits around the sun, to do so they considered only the two bodies: the mass and position of a single planet and the sun. The truth is more complex, because all masses exert gravity, and as soon as we introduce even a third mass there is no general closed-form solution and some systems are even chaotic. The first problem studied was the Sun-Earth-Moon 3-body problem, which Newton could not solve and succeeding generations continued to pursue. Early physicists became so frustrated that they began to doubt Newton's Law of Universal Gravitation. Renown mathematician Leonard Euler even wrongly argued against the inverse square law. Being able to accurately predict the Moon's orbit had huge implications for navigation and determination of longitude at sea. Competing polymaths Jean le Rond d'Alembert and Alexis Clairaut each presented their analyses of the problem to the Académie Royale des Sciences in 1747. Clairaut had found a brilliant approximate solution to the 3-body problem for which he received the 1750 prize of the St Petersburg Academy for his essay "Théorie de la lune". 

In 1757, Lalande decided he improve on the predictions of a different 3-body problem: the return of Halley's Comet, last seen in 1682. Because the gravitational pull of Jupiter and Saturn cannot be neglected, Halley himself was only able to calculate that the comet would return "around the end of the year 1758 or the beginning of the next." He enlisted the help of Clairaut and Lepaute.  The divided up the calculations required and worked in parallel for more than 6 months, barely even stopping to eat! They were in a race to make their prediction before the comet itself arrived. By November 1758 they gave a two-month window for comet's perihelion (closed point to the sun) of the 15th of March to the 15th of May, centered around the 13th of April 1759.  They missed the comet's arrival, the 13th of March, 1759 by only a couple of days. Sour grapes Jean d'Alembert griped that their work was "more laborious than deep". In fact, their heroic efforts were a huge technical feat and ten-fold improvement on Haley's vague two-year window. Their error was only due to employing the less-than-accurate accepted masses for Jupiter and Saturn. Notorious ladies' man Clairaut unfortunately removed any mention of Lepaute from his 1760, "Théorie du mouvement des comètes", alledgedly to please another woman, whereas in Lalande's "Théorie des Comètes" he insists they could never have made the calculations without her. 

Lepaute went on to collaborate with Lalande and his calculations for decades. In 1759 Lalande became the director of the  Académie des Sciences astronomical almanac Connaissance des Temps (Knowledge of the times) and appointed Lepaute as his assistant. He prepared computing plans and she did the calculations for the almanac. Her work included calculations on a 1762 comet, and a table of parallactic angles, work on the Éphémerides, annual guides for astronomers and navigators, calculating the daily position of Saturn from 1775 to 1784 (for the seventh volume, in 1774). She calculated on her own the daily positions of the Sun, Moon and planets for the eighth volume (in 1784).

In 1762 she calculated the exact time and path of the annular solar eclipse of 1st of April, 1764. Under her own name, she published a map which showed the eclipse's extent over Europe (shown in green in my print), as well as the its successive phases in 15-minutes intervals as would be visible over Paris (shown in blue in my print).

She and her husband were childless but adopted his nephew, Joseph Lepaute Dagelet in 1768 and she trained him as a mathematician and astronomer. He became a professor and was inducted in the French Royal Academy of Sciences in 1785.

Despite how vital, and Lalande's vocal appreciation, her work remained largely unrewarded and unrecognized during her lifetime. She worked as computer for Lalande for 15 years while he was a professor and director of the Paris Observatory. She did became a member of the distinguished Scientific Academy of Béziers in 1761, and calculated the ephemeris of the 1761 transit of Venus for them. Her eyesight after decades of calculation deteriorated to the point that she had to retire in 1783. She spent the end of her life caring for her terminally ill husband. After her death in 1788, Lalande wrote a biography of her contributions which he included in his Astronomical Bibliography. Both an asteroid (7720 Lepaute) and a lunar crater have posthumously been named in her honour.

For Balance, I made my portrait of Zhang Heng (see previous post).

For Texture, I made a print of an Eastern Snail Shell Mason Bee, Osmia conjuncta. 

Eastern Snail Shell Mason Bee, linocut 8" x 8" on Japanese kozo paper, by Ele Willoughby, 2023

The Packer Lab at York University posted an image of this mason bee, Osmia conduct, and explained that these adorable little blue snail shell nesting bees had now been observed in Canada (southern Ontario) and to be honest, I'm completely obsessed with the idea. I think it's the cutest thing I've ever heard. Apparently, they think these bees have become more common here because there are now so many of the Cepaea snails, introduced from Europe. So, while I couldn't find any images of this bee nesting, I illustrated it with a Cepaea shell. You can find images and video of other Osmia bees nesting in snail shells online. Many thanks to the friendly and helpful Entomology Twitter folks who helped me track down the right type of snail shell and even introduced me to researchers who said they would try to get video of these cuties next field season!

I will miss Science Twitter when it's gone.

References for scientist bios

Johannes Kepler, Wikipedia, accessed January 2023

Nicole-Reine Lepaute, Wikipedia, accessed January 2023

Jérôme Lalande, Wikipedia, accessed January 2023

Alexis Clairaut, Wikipedia, accessed January 20223

Lynn, W. T. (2 January 1911). "Madame Lepaute". The Observatory. 34: 77–78. Bibcode:1911Obs....34...87L. 

Bernardi, Gabriella (21 March 2016). The Unforgetten Sisters: Female Astronomers and Scientists before Catherine Herschel. Springer. pp. 121–127. ISBN 9783319261270.

De La Lande, Jérôme (1803). Bibliographie astronomique avec l'histoire de l'astronomie depuis 1871 jusqu'à 1802 (in French). Paris: Imprimerie de la République. ISBN 978-2329549170. Archived from the original on 4 May 2010. 

Connor, Elisabeth (November 1944). "Mme. LePaute, An Eighteenth Century Computer". Astronomical Society of the Pacific Leaflets. 4 (189): 314–321. Bibcode:1944ASPL....4..314C.