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Wednesday, June 8, 2022
These are a Few of Our Favourite Bees
Thursday, April 28, 2022
Virginia Ragsdale's Conjecture
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| Virginia Ragsdale, linocut, 11" x 14" on cream coloured washi, by Ele Willoughby, 2022 |
One of the unexpected delights of making a series of portraits of people in STEM is that people are interested in subjects because of their personal relationships. In hindsight, it should have been obvious; I am a scientist and have family and friends! But, it didn't occur to me that these "famous" (in some circle) great scientists would appeal as images of family members, or friends, and it's actually pretty touching when people contact me and tell me they are buying a print because the subject is someone they knew personally. This is a mathematician who was introduced to me by a repeat customer, because this is one of her ancestors. So not only to I get to learn of a new-to-me historic women in STEM, but I have a commission to make her portrait for someone's own personal connection to her.
This is a linocut portrait of mathematician Virginia Ragsdale (1870-1945) who is remembered for the Ragsdale conjecture (shown in the upper two inequalities). I'm always wary of including equations in talking about the history of math and science. What's important is that she proposed a way of approaching a difficult problem that was an inspiration for a lot of other mathematicians. The problem (which remains unsolved!) was included in David Hilbert's famous set of problems, namely, what are the possible arrangements of real algebraic curves embedded in the projective plane? She decided to set an upper bound for certain type of such curves. She proposed that they consider algebraic curve of degree 2k which are all topologically circles (or ovals) where some ovals are nested inside each other; others are not. An oval is defined as even if it is contained an an even number of other ovals of the curve, otherwise the oval is called odd. She conjectured that for an algebraic curve which contains p odd and n even ovals:
It was a very useful insight to consider even and odd ovals separately; the difference p-n is the Euler characteristic of a region bounded by the even and odd ovals. The Ragsdale conjecture, made in her 1906 dissertation, is amongst the earliest and most famous on the topology of real and algebraic curves, which stimulated a lot of 20th century research and was not disproved until 1979. A correct upper bound has yet to be found. I have also included an inequality she posed:
later proved by Ivan Petrovsky. The diagrams of algebraic curves also appeared in her dissertation "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. In it she tackles the 16th of David Hilbert’s famous 23 unsolved problems in mathematics - one of only a few which remain unresolved today.
Born in Jamestown, North Carolina just after the Civil War, she was class valedictorian at the Salem Academy, where she excelled at math and piano. She was went on to Guilford College where she helped set up a YMCA, and establish an Alumni Association as well and worked to expand college athletics. When she graduated with a B.S. in 1892, she won a scholarship to Bryn Mawr for the woman student with the highest grades, so she continued on, studying for her A.B. degree in physics. She won a fellowship to study in Europe for the class of 1896, which she delayed for a year, working as a physics demonstrator and beginning graduate studies in math. She then spent 1897-98 attending lectures by the renown mathematicians Felix Klein and David Hilbert at the University of Göttingen.
Upon returning the the US, she taught math for three years in Baltimore before another scholarship, awarded by the Baltimore Association for the Promotion of University Education of Women, allowed her to return to Bryn Mawr to complete her doctorate with Charlotte Scott.
She moved to New York and taught at Dr. Sach’s School for Girls until 1905. She became the head of the Baldwin School in Bryn Mawr from 1906-1911 and worked as Charlotte Scott’s reader from 1908-1910. She accepted a mathematics position which brought her back to North Carolina in 1911 at the Women’s College in Greensboro (now UNC at Greensboro) where she stayed for almost two decades. She was department head from 1926-1928 and left a lasting impact, insisting on investing in a telescope and adding statistics to the curriculum. She held high standards but was known for her patience for students.
She retired to care for her ailing mother and run the family farm in 1928. When her mother died she built a lovely house at the edge of the Guilford College where she gardened, restored furniture and researched her family’s genealogy. Upon her death she left the home to the college and serves now as the home of the college president.
References
Wednesday, April 6, 2022
Movements in Art History Inspired Prints
I haven't yet posted all the prints I made for #PrinterSolstice so here are some of the others!
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| Dada-inspired linocut by Ele Willoughby |
This print combines my spherical cow print, an eye and a skeleton with the repeated word "Dada" and is unique... but I made a second print with some of the same elements, but a tardigrade, rather than a skeleton.
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| Dada-inspired linocut by Ele Willoughby |
This is a minimalist print inspired by the transit of Venus. Much like a solar eclipse, when our moon's orbit brings it between the Earth and sun, the transit of Venus occurs when the planet Venus passes directly between the sun and us (or any other planet) partially obscuring or occulting the disk of the sun. Venus is much larger than our moon, but much farther away, so it blocks a much smaller portion of the sun, for a much longer period; it typically appears like a small black dot slowly moving in a line across the face of the sun over the course of several hours. These transits are rare, but predictable. The separation between transits is: 121.5 years, 8years, 105.5 years. There have been two this century (in 2004 and 2012) and there will not be another until 2117. Historically such transits have been vital to astronomers in their efforts to gauge the size of our solar system. Astronomers travelled to remote sites across the globe to get multiple simultaneous observations of the event, and improve distance and size estimates using parallax of the various observations.
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| The Transit of Venus, linocut by Ele Willoughby, 9" x11", 2022 |
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| Goethe's Theory of Colours, Linocut by Ele Willoughby, 8" x 10", 2022 |
Goethe rejected Newton’s description of colour and analytic approach. He wanted to portray rather than explain colour. He was a careful observer and claimed, “The human being himself, to the extent that he makes sound use of his senses, is the most exact physical apparatus that can exist.” He felt colour came from the interplay of light and dark through a turbid medium (like air, dust and moisture). He included aesthetic qualities such as the allegorical, symbolic, and mystic use of colour. His physics were not sound but he was the first to probe human colour perception and his insights inspired the art world (especially J. M. W. Turner) and philosopher Ludwig Wittgenstein to write ‘Remarks on Colour’.
My portrait is based on contemporary portraits, particularly a painting of Goethe in 1828 by Joseph Karl Stieler.
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| Horus and Seth, mono print by Ele Willoughby, 8" x 10", 2022 |
This is a one of a kind mono print about the Ancient Egyptian gods Horus and Seth, representing the struggle between order and chaos. My print is inspired by the Harlem Renaissance paintings of Aaron Douglas in particular, his layers of translucent colours, silhouettes and use of ancient Egyptian (and other African) culture, myth and imagery.
This print shows Horus battling his uncle Seth. The story of their fight is NSFW as they say, but what I like in particular is that the ancient Egyptians believed both the god Horus, who represented order, amongst other things, and Seth (or Set), who represented chaos were necessary components for life and creativity.
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| Flamboyant Wormhole, linocut by Ele Willoughby, |
This is a hand-printed Lino block print of an astrophysical wormhole, linking two locations in spacetime, inspired by Op Art. A wormhole, also called an Einstein-Rosen bridge, may exist, linking points in space-time like a sort of tunnel from one space and time to another. We don't know if they actually exist, but they are consistent with General Relativity and have been the subject of a lot of theoretical physics research and sci-fi. Space time is 4D, but if we represent it on the page, we can imagine it like a flat surface and then a wormhole would be a sort of tube from one area to another with a blackhole like a sort of drain leading to a white hole. If such structures exist, they may or may not be traversable, but it's conceivable they could allow time travel or faster than light travel or travel between different universes depending on the location of each end.
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| Theia, linocut print by Ele Willoughby, 8" x10", 2022 |
This 6 colour hand printed linocut print illustrates the formation of the Earth’s Moon in a comic book influenced pop art style. The favoured model of Moon formation is that 4.5 billion years ago, early in the history of our solar system, a Mars-sized astronomical body called Theia crashed into the early proto-Earth. Large amounts of projectiles from this catastrophic collision, both rocks from Theia and the proto-Earth’s own mantle, coalesced under gravity to form this planet’s own satellite: the Moon. The giant-collision theory is supported by moon rock samples gathered during the Apollo missions and lunar asteroids found on Earth; these rocks so similar to the Earth's chemical and isotopic make-up suggest that the Earth and Moon have some shared history and that the Moon was not simply a capture object from elsewhere. But variations in the types and proportions of minerals found in Moon rocks versus here on Earth suggest that there's more to the story and it was not simply formed simultaneously with the Earth. The Moon is rich in minerals formed at high temperatures like we would expect from a massive impact event.
We can thank this catastrophic event, sometimes called 'The Big Splash' for the world as we know it, with stable orbit and climate thanks to having such an unusually large satellite in our Moon.
Wednesday, February 2, 2022
As we warm the Arctic
"This is the Arctic with the range of the polar bear: its habitat. And this is what happens when we heat it.”
This is an interactive lino block print which changes colour! A map of the Arctic looking down on the North pole is handprinted in a delicate cream colour on white Japanese kozo (or mulberry) washi paper. On top of the map, I have printed the range of the polar bear in blue complete with a polar bear and the aurora (or Northern Lights) and the word habitat in fuchsia. I have mixed the blue and fuchsia inks myself: they are made with thermochromic temperature sensitive pigments. If these inks rise about 30 degrees Celsius (70 degrees Fahrenheit) they turn colourless, so if you heat the print, for instance with a hair dryer, the range, polar bear and word habitat all disappear. The print is called “As we warm the Arctic” and is about habitat loss due to climate change.
Thursday, January 27, 2022
Poincaré, Cubism, Non-Euclidian Geometry and Special Relativity
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| Cubist Poincaré, linocut, 8" x 8" by Ele Willoughby, 2022 |
This is a hand-printed lino block portrait in the Cubist style of mathematician Henri Poincaré. It’s one of a variable edition printed on Japanese kozo (or mulberry paper), 8” x 8” or (20.3 cm x 20.3 cm). The latest prompt for #printersolstice is Cubism.
Diagram from Jouffret's 'Traité élémentaire de géométrie à quatre dimensions'
This print is about how movements in art can be connected with contemporary math and physics (and other sciences). Specifically, the way Cubism breaks from a single favoured perspective or absolute frame of reference and attempts to break down subjects into geometrical shapes from multiple points of view can be tied to advancements in non-Euclidian geometry in mathematics, special relativity (published by Einstein in 1905) and the revolution in early 20th century physics. Picasso and his circle were specifically introduced to the mathematics of the fourth dimension and Non-Euclidian geometry by his actuary and enthusiastic amateur mathematician friend Maurice Princet, who gave him the book 'Traité élémentaire de géométrie à quatre dimensions' (Elementary Treatise on the Geometry of Four Dimensions) by Esprit Jouffret, a popularization of Henri Poincaré's 'La Science et l'Hypothèse' (Science and Hypothesis). Jouffret's book includes many diagrams of four dimensional objects like hypercubes projected on the 2D surface of a page. Princet also spoke to Picasso about the work of polymath Henri Poincaré. Picasso realized that this approach to 4D geometry captured the how he wanted to depict two points of view simultaneously and art historians can trace the influence of this geometry in his earliest Cubist paintings. So I have made a Cubist style portrait of mathematician, physicist, engineer and philosopher of science Henri Poincaré (1854-1912).
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| Henri Poincaré (photograph published in 1913, via Wikipedia) |
Pointcaré was called "The Last Universalist" since he excelled in such a wide range of mathematics including pure and applied math, mathematical physics and celestial mechanics. Chaos theory and the idea that deterministic systems can have sensitive dependance on initial conditions (colloquially known as the butterfly effect) seemed to explode on the scene with increasing applications in the 1980s but in fact some foundations were laid a century earlier by Poincaré. He was the first person to discover a deterministic system which exhibits chaos in the three-body problem (or how a system of 3 masses behave over under Newton's Law of Universal Gravitation). The three circular shapes in the background allude to the 3-body problem and the spiral is a hint at the chaos in the trajectories. He is considered a founder of the study of topology (how geometric properties are preserved under deformations) and stressed the importance of the invariance of physical laws under various transformations. Special relativity tells us that light speed is the universal speed limit, and that space and time are not absolute. As things approach this speed limit they experience length contraction and time dilation. We use Lorentz transformations to compare length and duration in differing frames of reference. It was Poincaré who wrote the Lorentz transformations in their modern form. He discovered the Lorentz transformations for velocity and the invariance of Maxwell's equations (for electricity, magnetism and light), which was vital part of special relativity. He published a short paper in 1905 which Einstein did not see until after he had published his own independent and revolutionary paper on special relativity. Poincaré also proposed gravitational waves, travelling at light speed from masses. It's hard to over-state his importance to 20th century math and physics.
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| Pablo Picasso, Portrait of Ambroise Vollard, spring of 1910 (Pushkin Museum, Moscow) |
Poincaré himself worked quite intuitively and wrote, "It is only through science and art that civilization is of value."
One of the ways in which Special relativity is strange is that we discovered the problem of simultaneity. If there is no absolute frame of reference, there is no simultaneity; observers in different inertial frames of reference (that is going different speeds*) will not agree on the sequence of events or even whether two things happen at the same time. This problem of simultaneity, it can be argued, is exactly what interested Picasso, when he chose to paint two different points of view on the same canvas. This is the argument made by Miller, author of 'Einstein, Picasso: Space, Time and the Beauty That Causes Havoc' makes. Leonard Schlain, in 'Art & Physics' makes the case that Picasso in painting all sides of an object simultaneously as if he was seeing space as all here in the everlasting now, was depicting what a viewer would see astride a beam of light at the universal speed limit when time has reached its maximum dilation to the present (and only the present).
Earlier art historians, and in fact Einstein when asked, argued that Cubism wasn't connected to relativity or non-Euclidian geometry but I think each under appreciated the other. The art historians argued the artists had no knowledge of Einstein, or Minkowski (who combined space and time into the 4D spacetime), equations or pure math. Einstein preferred earlier classical painting and argued "This new artistic 'language' has nothing in common with the Theory of Relativity" but then, he felt contemporary art and music had both degenerated. If we are less literal, I think the connection between the two movements is quite clear. There is no absolute frame of reference and both art and physics were revolutionized by this realization.
Schlain makes a further argument connecting Cubism with relativity but noting that colours also change with speed, as length contracts (including the wavelengths or colours we would observe), and colours would be merge as what is in front and behind become one. He argues we would be left with neutrals at the speed of light: white (of white light), black (of its absence), and muddy tones of mixed colours brown and grey. These are precisely the colours Picasso and Braque used in their early Cubist paintings, and the colours of my palette for my portrait of Poincaré.
References
William B. Ashworth, Jr., Scientist of the Day - Esprit Jouffret, Linda Hall Library blog, March 15, 2021.
Arthur I. Miller, Henri Poincaré the unlikely like between Einstein and Picasso, The Guardian, July 17, 2021
Did Picasso know about Einstein?, Physics World, November 1, 2002
Wednesday, January 12, 2022
Prints inspired by art history
I am trying to participate in the #PrinterSolstice again this year. They have created a series of prompts for each of 13 weeks starting with the (northern hemisphere) winter solstice. This year the themes are based on 20th century movements in art. It's been challenging to find time; we're once again sheltering at home and schools have gone remote here. Facilitating remote school takes a lot of my time and I cannot print at the same time, though I can plan, draw and carve.
We started with Abstract Expressionism, which is really outside my wheelhouse, since my prints are representational. But it gave me a chance to work intuitively to create a mono print with scraps of lino.
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| monoprint by Ele Willoughby, 2022 |
Next was Art Deco. I looked at a lot of Art Deco artworks. The frequent clamshell motif and the keyhole shape in Art Nouveau works, plus seeing a number of mermaid illustrations reminded me I had long wanted to illustrate the story of Mélusine, the legendary half-fairy ancestor of the House of Lusignan and royals including the Plantagenets, whose lower half became a serpent when she bathed. She marries a nobleman but insists he must never observe her in the bath; of course he does, and she leaves him - so the keyhole shape works just right to tell the story! There are many versions, like most folklore. Sometimes she has two tails, or a fishtail or wings. There are different versions of her name and various stories are found in different regions especially France, Luxembourg and the Low Countries. I first learned of her from Manuel Mujica Láinez’s The Wandering Unicorn, which tells her story through the centuries. Her story also appears in A.S. Byatt’s Possession amongst many other places from operas to video games.
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| Mélusine linocut, 8" x 10" by Ele Willoughby, 2022 |
These art history prompts have been a challenge to link to my art practice. I've been seeking a connection to things like science art. The third prompt was Bauhaus. In looking at Bauhaus paintings I was reminded of nothing so much as Feynman diagrams, and the exuberance action on the quantum scale, so I decided to combine quantum physics with the Bauhaus vocabulary of lines, shapes and colour palette. I was thinking more of the paintings of Kandinsky than the regular geometric shapes in Bauhaus design: a welter of black straight, wavy and curly lines, stripes of colour, triangles, circles and concentric rings and washes of pale colour, often on a cream coloured background.
The Feynman diagrams are a tool of particle and quantum physics to both denote a particle interaction and also can be used to make calculations of probabilities and physical properties. Straight lines denote particles (quarks and leptons like electrons). Wavy lines denote photons. Curly lines are gluons and the dotted line is a Higgs particle. I wanted to include an electron spontaneously giving off a virtual photo and particle-antiparticle pairs, production of a Higgs particle, one diagram of the sort of we see if we smash protons together, and one hilariously named penguin diagram (in this instance producing a gluon).
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| Feynman Diagram Bauhaus, linocut, variable edition, 8" x 10" by Ele Willoughby, 2022 |
Looking at all these prints as I try to plan ahead and link them to my art practice and science art I am reminded of Leonard Shlain's book Art and Physics. It should be around here somewhere but I can't find it on my bookshelves. (Aside: this house needs a librarian.) In the book Schlain linked major movements in art with advancements in physics. I am not sure I can agree with his thesis that movements in art presaged new physics (which feels like it violates causality to me) but I greatly enjoyed the book. Others have drawn parallels between art and physics and it's clear that both math and physics were an influence of some of these major modern movements in art. In fact, this has been true for centuries; consider for instance the development of artistic perspective, or the camera obscura. One of the obvious connections is between cubism and the math of the fourth dimension and the physics of relativity. So that is something I'm thinking about for an upcoming print.
Friday, December 3, 2021
Somewhat sinister holiday cards
I decided to add to my collection of darker yuletide folklore cards, to go with the Yule Cat and made:
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| linocut Krampus card by Ele Willoughby, 2021 |
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| linocut Mari Lwyd card by Ele Willoughby, 2021 |
The name Mari Lwyd (pronounced "lood") might come from Holy Mary or grey mare. The source of the tradition is debated but can be linked to various British hooded animal traditions.
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| linocut Joulupukki, or the Yule Goat, card by Ele Willoughby, 2021 |
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| linocut Perchta card by Ele Willoughby, 2021 |
Some legends associate her with the Wild Hunt, and claim she rides through the night sky with her Perchten.
In contemporary alpine festivals she is accompanied by an entourage, the Perchten, either beautiful and bright Schönperchten who bring luck or ugly Schiachperchten with fangs, tusks and horse tails (resembling Krampus) who are supposed to drive out ghosts and demons. She is viewed as the one who rewards generosity and punishes bad behaviour.