Friday, June 11, 2021

Imaginary Friends of Geology

Painting by Alexander Craig during the British Association meeting in Glasgow in 1840 when Lyell was 43 years old.
Charles Lyell, by Alexander Craig in 1840 when Lyell was 43 years old.
I’m always on the lookout for new “imaginary friends of science” and periodically I ask for suggestions. My series is made of the charismatic metaphors and thought experiments through the history of science. “Imaginary friend” is a bit tongue-in-cheek but my series includes Schrödinger’s cat, Maxwell’s Demon, Laplace’s Demon, Descartes’ Demon, Russell’s Teapot, Occam’s Razor, the Spherical Cow and Hilbert’s Grand Hotel. My series is skewed towards physics (with some math and philosophy) because that’s my background. I ask for suggestions and biologists just look quizzical but I suspect other fields do this too. It's not an obvious thing to research since this is not really a recognized category of metaphor/thought-experiment, and I coined the term "imaginary friend of science" but you sort of know one when you meet them. Tell me your favourites!

Maxwell's, Laplace's and Descartes' Demons

Recently I got a suggestion from geology for my ‘imaginary friends of science’ series (hat tip to Greg Priest). This imaginary friend is a bit obscure but fits perfectly. Early geologist Charles Lyell (1797-1875), in his famous ‘Principles of Geology’ wrote that to avoid some sources of prejudice in understanding geology would require an Amphibious Being, who could, say, compare processes happening today on land and those happening today below water with what we see in the geological record. Likewise the Amphibious Being would be better able to identify fossils, being familiar with life on land and in the sea. He wrote,

It should, therefore, be remembered, that the task imposed on those who study the earth's history requires no ordinary share of discretion; for we are precluded from collating the corresponding parts of the system of things as it exists now, and as it existed at former periods. If we were inhabitants of another element—if the great ocean were our domain, instead of the narrow limits of the land, our difficulties would be considerably lessened; while, on the other hand, there can be little doubt, although the reader may, perhaps, smile at the bare suggestion of such an idea, that an amphibious being, who should possess our faculties, would still more easily arrive at sound theoretical opinions in geology, since he might behold, on the one hand, the decomposition of rocks in the atmosphere, or the transportation of matter by running water; and, on the other, examine the deposition of sediment in the sea, and the imbedding of animal and vegetable remains in new strata. He might ascertain, by direct observation, the action of a mountain torrent, as well as of a marine current; might compare the products of volcanoes poured out upon the land with those ejected beneath the waters; and might mark, on the one hand, the growth of the forest, and, on the other, that of the coral reef. 

-Charles Lyell, 'Principles of Geology' 9th ed. 1853

James Hutton during fieldwork, by John Kay

He actually goes on to further propose other biases could be avoided only if one could live also in the subterranean world, and hence, a further imaginary friend, "some "dusky melancholy sprite," like Umbriel, who could "flit on sooty pinions to the central earth," but who was never permitted to "sully the fair face of light," and emerge into the regions of water and of air" but such a sprite would have biases due to his lack of access to life on land or in the sea! Here I'm just going to consider the Amphibious Being. 







The Geologist - Carl Spitzweg, circa 1860
The Geologist - Carl Spitzweg, circa 1860
While far less well-known than say, Maxwell's Demon or Schroedinger's Cat, the Amphibious Being absolutely performs the same purpose. He's a figment of Lyell's imagination who serves to elucidate and explore a problem in science. The Amphibious Being highlights the possibility that our conclusions about the origins of rocks or fossils  are biased since we only live on land, the same way Maxwell's Demon explores the mysterious nature of the Second Law of Thermodynamics or Schroedinger's Cat exposes the strangeness of the probabilistic nature of quantum mechanics and what that would mean if we saw it in the every day world of large things like well, cats, rather than the subatomic scale. So I plan to add the Amphibious Being to my collection.

Rise of Mr. de Saussure to the summit of Mont Blanc, aquarellée engraving of Christian of Mechel (1737- 1817).

Professor Owen, by Frederick Watty in Cartoon Portraits and Biographical Sketches of Men of the Day (1873)

My question is, what would such an Amphibious Being look like? I made my Demons (Maxwell's, Laplace's and Descartes') ressemble those who introduced them with some additional features like horns. There are a lot of images of Lyell at various ages. He is always wearing a suit, with a magnifying glass or monocle on a chain around his neck.  There are sadly no images of him in the field. I tried to find some images of early geologists or Victorian geologists in particular. It's easy to find depictions of James Hutton (1726-1797) in the field, but not Lyell. Even searching for Victorian geologists in general proved hard. The vast majority of paintings or later photographs show middle class men in suits at home. There are some hilariously heroic posed images of geologists with hammers looking like Classical heroes. I was pleased so see images of Mary Anning, Alice Wilson and other trail-blazing women in geology come up in my search. The Rise of Mr. de Saussure to the summit of Mont Blanc, an aquarellée engraving of Christian of Mechel (1737- 1817), shows that geologists of Lyell's time did in fact where suits in the field, and some of the simple tools they employed: walking sticks, rakes, shovels and packs of supplies. There are also a number of photographs of somewhat scruffier Gold Rush geologists in California or the Yukon. Along with simple tools, geological hammers, magnifying glasses, and collecting bags are common. I wondered also about Victorian caricatures? I have seen cariactures of various well-known historical scientists (even if scientists seem to rarely be the sort of recognizable public figure to appear in cartoons today). So I looked up cariatures of Charles Lyell and was not disappointed.

The caricature by De la Beches of Charles Lyell as Prof. Ichthyosaurus on the pages of Francis Trevelyan Buckland (Son of William B.). "Curiosities of Natural History".
'Awful Changes' - Caricature of Charles Lyell as Prof. Ichthyosaurus, by geologist Henry De la Beche (1796 -1855) on the pages of Francis Trevelyan Buckland (Son of William B.). "Curiosities of Natural History".

Ichthyosaurus caricature published in 1885 in the Punch magazine. The discovery of ichthyosaurus fossils lead to interest in reportings of sea monsters, including for Charles Lyell who wanted to support his idea that animals did not become extict, only rare. He wisely did not include this idea in his geology textbook. (via History of Geology)

The Professor Ichthyosaurus, though an actual, contemporary and amusing cariature of Lyell as a being, which in the cartoon at least, appears amphibious, isn't quite what I want. The itchyosaurus was a marine reptile, not an amphibian. The lecture setting does not suggest fieldwork, nor any greater insight into geology because of its amphibious nature. 

Henry de la Beche, incidentally, famously painted the first paleoart scene, Duria Antiquior, based largely on his friend Mary Anning's findings and he sold lithographic prints of his painting to raise funds to help support her. I thought of him as more modern and kind, in his loyalty and friendship with Mary Anning, a paleontologist who was largely excluded due to her sex, class and religion. But in researching this my opinion of him fell. While he understood slavery was an abomination, since his own income depended on an enheirtance of a sugar plantation in Jamaica worked by enslaved people, he (guiltily) opposed abolition in favour of in favour of gradual change. 'This is wrong but let's not be too hasty' is not an ethical stance.

Duria Antiquior – A more Ancient Dorset is a watercolour painted in 1830 by the geologist Henry De la Beche

And what about the all-important "amphibious" part of my Amphibious Being? My first thought was salmanders. I love the look of gill-stalks and perhaps they suggested Lyell's substantial sideburns. I find the axolotl looks quite anthropomorphic as it is (as reflected in Julio Cortazar's short story, Axolotl).  The axolotl is of course limited in range to Mexico, and though other salamanders go through a larval stage with gill-stalks, this didn't feel quite right or in any way an allusion to Lyell. My next thought was to look at Victorian cariatures of anthropomorphic frogs, which (for reasons I don't entirely understand) was very much a thing. Frog men cariatures and Christmas cards and other illustrations, sometimes to satirize society are pretty common, so I thought that was just the thing.

I am now sketching a Victorian frog-man geologist for my next Imaginary Friends of Science print!

Wednesday, June 9, 2021

Hilbert's Grand Hotel


Hilbert's Grand Hotel, 8" x 10", linocut by Ele Willoughby, 2021

My latest linocut print illustrates the paradox of Hilbert's Grand Hotel, a thought experiment conceived by the great mathematician David Hilbert (1862-1943) in 1924 to show the paradox of infinite sets. If you imagine a regular hotel with a finite number of rooms, if they are all filled you can't add more guests. In an imaginary infinite hotel however, guests are always welcome! Even if all the (conveniently numbered 1, 2, 3, ...) rooms are filled, you simply shuffle all the guests over one room and make room for the next guest. You can repeat this argument for one more guest, two more guests, three more guests, ...infinity more guest... infinity groups of infinite guests, etcetera!

The print shows a grand hotel on a steep angle (with a vanishing point at infinity of course), labelled "Hilbert's Grand Hotel" below a cameo of Hilbert himself, and the slogan "No vancancy. Check in now!" Above the hotel is a glowing infinity symbol. The size of countably infinite sets, like all natural numbers {1,2,3, .... ∞ } is denoted by Aleph Nought ℵ0 shown on the two flags.

Tuesday, May 11, 2021

Botanist E.K.Janaki Ammal

E.K. Janaki Ammal
E.K. Janaki Ammal, linocut by Ele Willoughby, 9.25" x 12.5", 2021

Botanist Janaki Ammal (4 November 1897 – 7 February 1984), a trailblazer for women in science in India, worked on genetic crosses, breeding sweeter sugarcane varieties which could thrive in India, was an expert in cytogenetics and phytogeography and coauthor of the influential Chromosomal Atlas of Plants. The first woman in India to earn a doctorate and the first woman in the US to earn a doctorate in botany, she became an environmental activist and wrote about the importance of incorporating indigenous knowledge when working toward sustainable development. Her name lives on in the names of plants like the Magnolia kobus Janaki Ammal and the eggplant (known as brinjal or brengal in India) Janaki Brengal, and the recently developed delicate yellow rose named in her honour.

One of 13 siblings born to Dewan Bahadur E. K. Krishnan, a sub-judge in Madras Presidency who wrote books about local birds, and his second wife Devi Krishnan, in a blended family of 19 with 6 children from his first marriage, Janaki grew up in a home with many books on nature and wildlife. She was nonetheless born into the Thiyya caste, which was considered backwards, and this later affected her treatment as a scientist. She went to a missionary school in Tellichery before studying for her botany degree at Queen Mary College in Madras (now Chennai) and graduating with an Honours degree from Presidency College in 1921, where she developed a love of cytogenetics. She took a job teaching Women’s Christian College. Granted a prestigious Barbour Scholarship at the University of Michigan, she was able to avoid a marriage to a cousin, having watched so many sisters have arranged marriages, and come to the US for graduate school. Her niece recounted that she was detained at Ellis Island until she was mistaken for an Indian princess dressed in her traditional silks, with long dark hair, and she told her niece, “I did not deny it.” After her masters in 1925, she returned to teaching at the WCC, and then complete her doctorate as a Barbour Fellow in 1931, becoming the first woman in the US to earn a doctorate in botany. As a plant cytologist she focused on hybridization on plants, producing interspecific and intergeneric hybrids. Her thesis "Chromosome Studies in Nicandra Physaloides" saw her produce a triploid eggplant cross known as "Janaki Brengal."

She returned to India as a professor of botany at the Maharaja's College of Science, Trivandrum, from 1932 - 1934 before spending five years at the Imperial Sugar Cane Institute, Coimbatore along with Charles Alfred Barber, as a geneticist working to develop a sweeter sugarcane hybrid which could thrive in India and reduce their dependence on imports from Indonesia and grow more food domestically. She worked on the cytogenetics of Saccharum spontaneum, the sugarcane species grown in India, as well as developing hybrids with related grass species like zea, sorghum, and bamboo. She was elected Fellow of the Indian Academy of Sciences in 1935. Her cross-breeding research, using a labour intense process called polyploidy, provided the means for consistent results in hardy sugarcane with higher sucrose content. Her work on hybrid plants which could survive the tropical climate was important to India's recovery from famines (including the 1943 Bengal famine that claimed nearly 3 million lives).

Frustrated by discrimination she faced due to her sex, single status and her class, she moved to England in 1940 where she worked as Assistant Cytologist at the John Innes Horticultural Institution in London, throughout the Blitz, until 1945. For company, she smuggled a palm squirrel named Kapok in the folds of her sari, into the UK and it lived at the John Innes Horticultural Institution for many years! She met and worked with many celebrated geneticists (and eugenicists) there. She had been invited to work with famous geneticist Cyril Dean Darlington, and together wrote the monumental Chromosome Atlas of Cultivated Plants, which remains an important botany text on economic plants today. They took a different approach; rather than mapping the occurrence of plants of different classifications, their atlas recorded the chromosome numbers of 100,000 plants to show evolutionary patterns of botanical groups. She also published chromosonal counts, ploidy and on the origin and evolution of numerous garden species including grasses, Rhododendron and Nerines. S.D. Darlington and the renown J.B.S. Haldane (who coined the term ‘genetics’) remained friends and mentors throughout her career.

Then she worked as a cytologist at the Royal Horticultural Society at Wisley from 1945 to 1951; she was their first salaried female staff member. She researched the effects of the medication colchicine on woody plants like magnolia. When applied in water to the growing tip of young seedlings once their seed leaves had fully expanded it caused a doubling of chromosomes in cells, giving them heavier leaves, variable flowers with often thicker tepals which make them more durable. She planted many magnolia shrubs which still bloom every spring and a variety (with white petals and purple stamens) has been named Magnolia kobus Janaki Ammal after her.

After a chance meeting on a plane, none other than Prime Minister Jawaharlal Nehru lured her back to India post-Independence by personally requesting she return in 1951 to be Officer on Special Duty then later Director General of the Botanical Survey of India, which had been a colonial institution set up by Kew Gardens in 1890 to gather specimen (and export them to England). She found the colonialist European-biased tradition continued and she worked to have house Indian plant specimens in India. She wrote, “The plants collected in India during the last thirty years have been chiefly by foreign botanists and often sponsored by institutions outside India. They are now found in various gardens and herbaria in Europe, so that modern research on the flora of India can be conducted more intensely outside India than within this country.” She worked to document indigenous knowledge of plants and their cultural role. She argued that the way Chinese and Malayan flora mixed with Indian plants in north-east India led to natural hybridisation and resulted in great biodiversity. In 1956 the University of Michigan gave her an honorary LL.D. Nobel laureate C. V. Raman founded the Indian National Academy of Sciences and selected her as fellow in its first year in 1957.  She worked the rest of her career as a government scientist in various roles within India such as at Central Botanical Laboratory at Allahabad, as officer on special duty at the Regional Research Laboratory in Jammu, and at the Bhabha Atomic Research Centre at Trombay before settling in Madras in 1970 as Emeritus Scientist at the Centre for Advanced Study in Botany, University of Madras. She lived and worked in their Field Laboratory at Maduravoyal near Madras for the rest of her life and made her own garden of medicinal plants. She was the first woman scientist to have the prestigious Padma Shri conferred on her by the Government of India in 1977. During her retirement years, she focused on researching and publishing on medicinal plants and ethnobotany, travelling the country and interacting with remote tribes and making tremendous contributions to research on Northeast and Himilayan flora.

Her focus shifted to protecting the environment and biodiversity for the future. The push for food cultivation was leaving to massive deforestation, including 25 million acres claimed by the government under the Grow More Food Campaign. She used her renown as a leading scientist to defend the rainforest in Kerala with the Save Silent Valley campaign when the government planned to flood 8.3 square kilometres for a hydroelectric project. At 80 years old she used all her prestige to protect this pristine rainforest and all the biodiversity it contains, as well as putting her expertise to work mounting a chromosonal survey of all the plants it contains. The campaign was successful and the area became instead a national park, after her death at age 87 in 1984.  

Born under British rule, to a caste deemed backwards, when only 1% of Indian women were literate, and less than 1000 advanced beyond 10th grade, Janaki Ammal rose to prominence as a scientist. She spent years working in scientific institutions which were otherwise white male enclaves. She lead an ascetic life of Ghandian simplicity, always dressed simply in saris, wishing to be remembered for her scientific work at the cutting edge of cytogenetics and cytogeography.  She was a trailblazer for applying indigenous knowledge in environmentalism. Her name lives on in plant varieties, and the Ministry of Environment and Forestry of the Government of India National Award of Taxonomy, and several scholarships but her work lives on in the great swathes of rainforest she helped preserve, the sweetness of Indian sugarcane hybrids, hardy magnolias still living in England and much more.


Janaki Ammal, wikipedia, accessed April, 2021 

Sharanya Dutta, Do You Know the Botanist Janaki Ammal, She of the Magnolia Kobus fame?, The Wire, October 21, 2016

Archana Nagarajan, Janaki Ammal, Sci-illustrate Stories, July 7, 2019.

Janaki Ammal Edvaleth Kakkat, University of Michigan Rackham Graduate School Barbour Scholars Spotlight, accessed May 2021.

Janaki Ammal, History of Scientific Women on, accessed May, 2021.

Vinita Damodaran, Janaki Ammal - My work is what will survive, PHYTOPIA Science Gallery Bengalaru on, accessed May, 2021. 

Leila McNeill, The Pioneering Female Botanist Who Sweetened a Nation and Saved a Valley,, July 31, 2019.

Geeta Doctor, Celebrating Janaki Ammal, Botanist and a Passionate Wanderer of Many Worlds, The Wire, July 6, 2016

Mandeep Matharu, Yvette Harvey & Matthew Biggs. "My Work is What Will Survice". NatSCA blog. April 26, 2019.

Wednesday, April 7, 2021

Biochemist Marie Maynard Daly

Marie Maynard Daly linocut
Marie Maynard Daly, linocut 9.25" x 12.5" by Ele Willoughby, 2021

This is a hand-carved and hand-printed linocut portrait of trailblazing American biochemist Marie Maynard Daly (1921-2003). Daly, the first Black woman to earn a doctorate in chemistry in the US, made important research contributions to our understanding of the biochemisty of the cell nucleus and cardiovascular issues. Her interest in the nuclear proteins within cells lead to important contributions to our knowledge of the chemistry of histones and protein synthesis. She published original research establishing that  "no bases other than adenine, guanine, thymine, and cytosine were present in appreciable amounts" in DNA - research which was cited when Watson and Crick accepted the Nobel Prize for the structure of DNA. She did some of the earliest work on the relationship between diet and cardiovascular health. She was the first to show how cholesterol could clog arteries and that hypertension lead to atherosclerosis; these were invaluable discoveries in our understanding of heart attacks and work to lower the risk of heart attacks. She also did early work linking smoking and hypertension. Later she made studies of the uptake of creatine by muscle cells, which is important to understanding the recycling systems of muscles. She did this at a time when there where tremendous barriers due to race and gender discrimination in fields of research where women and minorities are still underrepresented.

Born April 16, 1921, in Queens, NY, she was the eldest of three and had twin younger brothers. Her father Ivan Daly was an imigrant from the British West Indies who had come to the US on a scolarship to Cornell to pursue his own dream of a career in chemistry, but when he and his family were unable to cover the tuition and board, he had to drop out of school. He supported his family as a postal worker, and passed on his love of science to his children. Her mother Helen Page Daly was a homemaker, avid reader and fellow lover of science, who also fostered her children's education. Marie's grandparents had a library with many biographies of scientists. Marie recalled being fascinated as a child by Paul DeKruips Microbe Hunters, with its stories of scientists like van Leeuwenhoek, Pasteur and Koch and being encouraged in her interests by parents and teachers alike. After attending an all-girls high school in Manhattan, she won a scholarship to Queens College.  She could remain close to home, and save money living at home. She graduated magna cum laude with a BSc in chemistry in 1942, in the top 2.5% of students, a Queens College Scholar and inducted into the Phi Beta Kappa and Sigma Xi honor societies.

She got lab and teaching assistant jobs at Queens College to support herself as she continued on, studying for her Master's at New York Univerysty, and afterwards. As WWII raged, and employment opportunities for a Black woman in chemistry were uncertain, she decided her best bet was to continue her education, and she began doctoral studies in biochemistry at Columbia under Dr. Mary Letitia Caldwell, Columbia's first female chemistry professor, a well-known expert in enzymes and nutritional chemistry. She graduated only three years after enrolling, in 1947, with her thesis "A Study of the Products Formed by the Action of Pancreatic Amylase on Corn Starch," unaware that she had become the first Black woman to earn a chemistry PhD, at a time when on 2% of Black women had college degrees. She was the first Black person to earn a doctorate from Columbia.

She wanted to work with biochemist Dr. Alfred Ezra Mirsky (one of the first to isolate mammal messenger RNA) but he told her she would need to provide her own funding. So she worked as a physical science instructor at Howard University for a year and began studying protein structure, until able to secure funding for a seven-year post-doctoral fellowship from the American Cancer Society. She moved to the Rockefeller Institute of Medicine in New York City, for the next seven years,  where she was the only Black scientist and she got to work with distinguished scientists. With Mirsky, she worked on protein synthesis and the metabolism of cellular nuclei, critical and foundational research for developing ideas on the structure of DNA.

She became  a biochemist associate for the College of Physicians and Surgeons at Columbia, and at the Albert Einstein College of Medicine in1955. There, she taught and continued her research on cancer and nucleic acids while she collaborated with Dr. Quentin Deming at Goldwater Memorial Hospital, on the causes of heart attacks. Her research about arterial walls looked at the effects of aging, hypertension, artherosclorosis (or the build up of fat in arteries), sugar and cholestorol. They showed that hypertension was a precusor to artherosclorosis and were the first to link cholestoral and clogged arteries. In studies of hypertensive rats, she showed the link between  high cholesterol and heart attacks. This was the foundation of work to prevent heart attacks. She studied the link between smoking and lung disease and how kidneys affect metabolism. She was a researcher for the American Heart Association from 1958 to 1963, working particularly on stroke research. Columbia made her an assistant professor of biochemistry from 1960-1961. She married Vincent Clark in 1961 (and retained her name professionally).

She and Deming both moved to Albert Einstein College of Medicine at Yeshiva University where she became an assistant professor of biochemisty and medicine (then associate professor from 1971). While at there, she started and helped run the Martin Luther King Jr. - Robert F. Kennedy program to help prepare Black students for admission and worked to recruit Black and Puerto Rican students. She also did important work on understanding the uptake of creatine in muscles, including the temperature and ion conditions it could be best absorbed. She was a dedicated mentor. She participated in a conference with 30 other minority women in STEM, hosted by the American Association for the Advancement of Science which resulted in in a report called The Double Bind: The Price of Being a Minority Woman in Science (Malcom, Hall, & Brown, 1976).

She was also a cancer scientist with the Heath Research Council of New York from 1962 through 1972. She served on the board of governors of the New York Academy of Science form 1974 to 1974. She was also active in student recruitment and in professional societies and the NAACP and National Association of Negro Business and Professional Women. She was a Fellow of the American Association for the Advancement of Science.

After retirement in 1986, she donated money to Queens College to create scolarships from Black students in physics and chemistry, in honour of her father. She served on the Commision for Science and Technology for New York City for three years, before moving with her husband Vincent Clark to their East Hampton summer home and then to Sarasota, Florida. In 1999, the National Technical Association recognized Daly as one of the top 50 women in STEM. She was devoted to playing the flute, gardening and her dogs; when her cancer made flute playing difficult, she learned the guitar. She died in 2003, at the age of 82, in New York City.


Daly, Marie Maynard., accessed April, 2021. 

Galen Scott, Black History Month - Marie Daly, The Researchers Gateway, March 7 2019

Marie M. Daly PhD Memorial Celebration, Einstein University, 2021

Victoria Corless, Pioneers in Science: Marie Daly, Advanced Science News, August 27, 2020 

Marie Maynard Daly, Wikipedia, accessed April, 2021.

Jalen Borne, Hidden Figures Beyond: The First Black PhD in Chemistry, Marie Maynard Daly, Charged Magazine, April 3, 2020

Dale DeBlakcsy, Marie Maynard Daly (1921-2003), America's First Black Woman Chemist, Women You Should Know, February 28, 2018

Daniel Tyrell, #BlackCardioInHistory: Dr. Marie Maynard Daly,, October 20, 2020.

Marie Maynard Daly, Science History Institute, November 9, 2018

Roopati Chaudhary, Marie Maynard Daly, Sci-illustrate, July 14, 2020.

Thursday, April 1, 2021

Chomsky Hierarchies, where math meets linguistics and computer language


Linocut 'Chomsky Hierarchy' by Ele Willoughby, 2021

This is one of the #mathyear prompts I had previously skipped. Chomsky hierarchies are the place math meets computer language and linguistics, and is something I have never studied, so I didn't have a good idea how to protray them. This year, I'm trying to do all the prompts I previously skipped, and I had a visiual idea.

This is a linocut portrait of linguist Noam Chomsky with a Venn diagram illustrating  the Chomsky hierarchy (or Chomsky-Schützenberger hierarchy) for formal grammars in formal language theory. In 1956, Noam Chomsky described the hierarchy of formal grammars (how to form strings from a language's alphabet that are valid according to the language's syntax). These ideas and hierarchy are pertinent for applied mathematics, theoretical computer science, theoretical linguistics and other areas.

He categoried 4 types of grammars, shown in the Venn diagram behind his portrait. Type 0, in pale yellow, contains recursively enumerable grammars, such that there exists a Turing machine which will enumerate all valid strings of the language. Type 1, in umber, contains context-sensitive grammars, are recognized by non-deterministic linear-bounded automata. Type 2, in orange, are context-free grammars, recognized by non-deterministic pushdown automaton. Type 3, in pink, are regular grammars, recognized by finite state automata.

Tuesday, March 16, 2021

Mathematician Sophie Germain


Linocut 'Sophie Germain' by Ele Willoughby, 2021
Linocut 'Sophie Germain' by Ele Willoughby, 2021

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


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


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


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


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


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


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


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


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


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

  1. if xp + yp + zp ≡ 0 (mod P), then P divides xyz, and
  2. p is not a p-th power residue (mod P).

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

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


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


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


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



Sophie Germain, Wikipedia, accessed March 2021

Sophie Germain’s Theorem, Wikipedia, accessed March 2021

Ernst Chladni, Wikipedia, accessed March 2021

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

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

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

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

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

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

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

Tuesday, March 9, 2021

Dmitri Mendeleev and the Periodic Table

Dmitri Mendeleev and the Periodic Table by Ele Willoughby
Dmitri Mendeleev and the Periodic Table, linocut by Ele Willoughby, 2021

Here’s my Dmitri Mendeleev block print for the Printer Solstice prompt “elements”. I had meant to make his portrait for the 150th anniversary of the periodic table in 2019, but I didn’t get to it. I couldn’t quite figure out how to indicate what he did, in contrast to our modern periodic table. So many people would recognize the shape of the periodic table, from high school, even if they aren’t scientists who use it regularly. But Mendeleev didn’t publish his idea in a form that’s easy to recognize.

First, his table, as published in 1869, is rotated by 90° so it shows groups in rows rather than columns. Second, prior to the discovery of protons and neutrons, he listed and organized elements by atomic weight (now called atomic mass), rather than atomic number (or number of protons). Third, some of his data wasn’t great, so sometimes elements appeared to have the same mass, or were out of order or even were mixtures. He lists Didymium, which is actually a mixture of the elements Praseodymium & Neodymium.

Sometimes you need to think about visual ideas for a long time. Rather than including his notes, or published results as well as a modern periodic table, my idea is to show how much of the modern periodic table he was able to deduce despite limited data. The elements that were unknown or unmeasured are blank- something the viewer can rapidly understand. In several cases he predicted we would find the missing elements in groups (columns). Then, while an impressive amount of the elements known in 1869 are exactly in the right place, there are several which he placed in the wrong groups (due to inaccurate masses). I plan to print those he grouped incorrectly in a different colour. I think this can give an immediate sense of 3 categories: elements he’d figured out, elements he hadn’t quite got right yet, & gaps as of yet unfilled. And if you’ve studied chemistry you can get a sense of what he figured out & why. Even when he placed elements in the wrong group he usual correctly deduced similarities in behaviour found within groups (columns) which we eventually figured out could be explained by structure at an atomic scale.

You can see from Mendeleev's 1869 publication, he got most of the known elements in the right sequence and many in the correct groups. He understood there were connections in properties in adjacent elements and periodicities of properties of elements in the same groups (which he wrote as rows and we now show as columns). He also corrently inferred several as of yet undiscovered elements (see the question marks).

 Mendeleev was born in 1834, in the village of Verkhnie Aremzyani, near Tobolsk in Siberia, the youngest of child of a large family. He was likely the 17th (though 3 older siblings died as infants and there is some dispute among sources). His father was a school principal until he lost his sight and his job. His mother then restarted her family's abandoned glass factory to support the family. His father died and the glass factory was destroyed by fire. Despite economic hardship, 13 year old Mendeleev attended the Gynasium in Tobolsk. In 1849, his mother took him all the way to Moscow to try to get into the university, they were unsuccessful. The now poor Mendeleev family moved to Saint Petersburgh in 1850 so he could instead attend the Main Pedagogical Institute. He graduated, but contracted tuberculosis and went to the Crimean to recover, where he became a science master of the 1st Simferopol Gymnasium. When he was recovered in 1857 he returned to Saint Petersburg. He worked on  capillarity of liquids and the workings of the spectroscope, published the textbook 'Organic Chemistry' and won the Demidov Prize of the Petersburg Academy of Sciences. He married  Feozva Nikitichna Leshcheva (1862), professor at the Saint Petersburg Technological Institute (1864) and Saint Petersburg State University (1865). He got his doctorate on "On the Combinations of Water with Alcohol" in 1865 and got tenure in 1867. He wrote the definitive two-volume chemistry textbook of his day, 'Principles of Chemistry'. By 1871, he had made Saint Petersburgh and international recognized centre of chemical research.

While working on his textbook, he was struck by the periodicity of properties. There had been some earlier, not quite successful attempts to organize elements by properties (of which he was not aware). He claimed to see it in a dream,

"I saw in a dream a table where all elements fell into place as required. Awakening, I immediately wrote it down on a piece of paper, only in one place did a correction later seem necessary."
— Mendeleev, as quoted by Inostrantzev

He started with 9 elements, 3 groups of 3 types of properties, then added the other known elements around the core of the table. On 6 March 1869, he presented  ' The Dependence between the Properties of the Atomic Weights of the Elements' to the Russian Chemical Society, using both atomic weight (now called relative atomic mass) and valence. He published his table in a a Russian language journal.

Modern Periodic Table

He correctly noted that if you arrange elements by their atomic mass there show repeating periodic properties. (We now know that atomic number is more important than atomic mass, but they usual would give you the same sequence, especially for lighter elements). He noted similaries in elements of similar atomic weights (that is, adjacent on the table), and similaries in elements in regularly increasing increments (that is, now in the same columns). He realized the elements were ordered by their valencies. He noted that there are more lighter elements. He noted that atomic weight determines properties (though we would now say atomic number). He predicted as-of-yet undiscovered elements at gaps in his table. He inferred that some of the atomic weight data was not quite accurate because the placement in the table did not line up with properties. He noted that knowing the atomic weight could help you predict an element's properties.

He met Anna Ivanova Popova, and divorced his wife in order to marry her, but the divorce was not finalized until a month after his wedding; further the Russian Orthodox Church stipulated 7 years were required between marriages; so, he caused a scandal in 1882. This is likely why he was never admitted to the Russian Academy of Sciences. But he received international acclaim, including receiving the Davy Medal (1882) and Copley Medal (1905) from the Royal Society of London. He resigned from Saint Petersburg University in 1890. He was elected a Foreign Member of the Royal Society (ForMemRS) in 1892. In 1893 he was appointed director of the Bureau of Weights and Measures, for the remainder of his life. He died in at 72, in 1907, from influenza.