Virginia Ragsdale's Conjecture

Virginia Ragsdale, linocut, 11" x 14" on cream coloured washi, by Ele Willoughby, 2022

One of the unexpected delights of making a series of portraits of people in STEM is that people are interested in subjects because of their personal relationships. In hindsight, it should have been obvious; I am a scientist and have family and friends! But, it didn't occur to me that these "famous" (in some circle) great scientists would appeal as images of family members, or friends, and it's actually pretty touching when people contact me and tell me they are buying a print because the subject is someone they knew personally. This is a mathematician who was introduced to me by a repeat customer, because this is one of her ancestors. So not only to I get to learn of a new-to-me historic women in STEM, but I have a commission to make her portrait for someone's own personal connection to her.

This is a linocut portrait of mathematician Virginia Ragsdale (1870-1945) who is remembered for the Ragsdale conjecture (shown in the upper two inequalities). I'm always wary of including equations in talking about the history of math and science. What's important is that she proposed a way of approaching a difficult problem that was an inspiration for a lot of other mathematicians. The problem (which remains unsolved!) was included in David Hilbert's famous set of problems, namely, what are the possible arrangements of real algebraic curves embedded in the projective plane? She decided to set an upper bound for certain type of such curves. She proposed that they consider algebraic curve of degree 2k which are all topologically circles (or ovals) where some ovals are nested inside each other; others are not. An oval is defined as even if it is contained an an even number of other ovals of the curve, otherwise the oval is called odd. She conjectured that for an algebraic curve which contains p odd and n even ovals:

 

and

.

It was a very useful insight to consider even and odd ovals separately; the difference p-n is the Euler characteristic of a region bounded by the even and odd ovals. The Ragsdale conjecture, made in her 1906 dissertation, is amongst the earliest and most famous on the topology of real and algebraic curves, which stimulated a lot of 20th century research and was not disproved until 1979. A correct upper bound has yet to be found. I have also included an inequality she posed:

later proved by Ivan Petrovsky. The diagrams of algebraic curves also appeared in her dissertation "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. In it she tackles the 16th of David Hilbert’s famous 23 unsolved problems in mathematics - one of only a few which remain unresolved today. 

Born in Jamestown, North Carolina just after the Civil War, she was class valedictorian at the Salem Academy, where she excelled at math and piano. She was went on to Guilford College where she helped set up a YMCA, and establish an Alumni Association as well and worked to expand college athletics. When she graduated with a B.S. in 1892, she won a scholarship to Bryn Mawr for the woman student with the highest grades, so she continued on, studying for her A.B. degree in physics. She won a fellowship to study in Europe for the class of 1896, which she delayed for a year, working as a physics demonstrator and beginning graduate studies in math. She then spent 1897-98 attending lectures by the renown mathematicians Felix Klein and David Hilbert at the University of Göttingen.

Upon returning the the US, she taught math for three years in Baltimore before another scholarship, awarded by the Baltimore Association for the Promotion of University Education of Women, allowed her to return to Bryn Mawr to complete her doctorate with Charlotte Scott. 

She moved to New York and taught at Dr. Sach’s School for Girls until 1905. She became the head of the Baldwin School in Bryn Mawr from 1906-1911 and worked as Charlotte Scott’s reader from 1908-1910. She accepted a mathematics position which brought her back to North Carolina in 1911 at the Women’s College in Greensboro (now UNC at Greensboro) where she stayed for almost two decades. She was department head from 1926-1928 and left a lasting impact, insisting on investing in a telescope and adding statistics to the curriculum. She held high standards but was known for her patience for students.

She retired to care for her ailing mother and run the family farm in 1928. When her mother died she built a lovely house at the edge of the Guilford College where she gardened, restored furniture and researched her family’s genealogy. Upon her death she left the home to the college and serves now as the home of the college president. 

References

Virginia Ragsdale, On the Arrangement of the Real Branches of Plane Algebraic Curves, American Journal of Mathematics, Volume 28, 1906

Virginia Ragsdale, Wikipedia, accessed April 2022

Ragsdale conjecture, Wikipedia, accessed April 2022

De Loera, Jesús; Wicklin, Frederick J. "Biographies of Women in Mathematics: Virginia Ragsdale". Anges Scott College. Retrieved April, 2022.

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!

Dada-inspired linocut by Ele Willoughby

This one of a kind lino block print combines multiple lino blocks in a Dada-inspired original artwork. Dada was a cultural movement post-WWI in response to the horrors of the war which often was satirical or nonsensical, breaking down all previous standards. Combining unrelated elements was common, as was the appearance of text, including of course, the repeated use of the word "Dada". It was a precursor to surrealism.

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.

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. 

The Transit of Venus, linocut by Ele Willoughby, 9" x11", 2022

The design of this print is inspired by Minimalism, an art movement post-WWII which espoused geometrical abstraction, and cutting away extraneous details, and sometimes even colours in favour of monochrome compositions. A hemicircle represents the solar disk and the series of smaller excised circles represent the path of Venus.

Goethe's Theory of Colours, Linocut by Ele Willoughby, 8" x 10", 2022

This print, inspired by the boldly coloured Fauvism paintings, is a hand-printed, limited edition, reduction block print portrait of writer, statesman and polymath Johann Wolfgang von Goethe (1749-1832). While best known for his contributions to German literature, he was pursued natural sciences, studying morphology, did work which presaged evolution, discovered the human intermaxillary bone, performed Galvanic experiments, studied anatomy with Alexander von Humboldt, formulated a theory of plant metamorphosis, gathered the largest private collection of minerals in Europe, and popularized a barometer he built based on Torricelli’s principle. What he considered his most important work was his Theory of Colours (Farbenlehre, 1810). 

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.

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.

This print was the first one I made with my new gel plate and stencils!

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.  

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.