When asked to contribute a Chronicle about their experience in writing, illustrating and revising the Notices article, in the August 2022 issue, the authors balked at first — didn’t the article speak for itself? Then Zizai Cue and Alex Winn, two students associated with the department of Mathematics at Duke University, proposed a list of questions for a Q&A Chronicle.
Here are the answers by the three authors, Susan Goldstine, Elizabeth Paley, and Henry Segerman, and by Bronna Butler, the artist who drew the map requested by the reviewers.
It is fascinating to learn more about the story of how mathemalchemy came about. How is doing mathematics similar to doing arts? How can the audience learn more about the two processes?
Research mathematics is often not that different from a kind of formal, logical poetry. Just as different poets may have different styles and ways of writing poetry, different mathematicians have different styles and ways of writing proofs, although just as in poetry, there are some standards and rules that everyone generally agrees on. Artists collectively explore the space of possible works, finding new areas and learning new techniques from each other. Mathematicians do the same.
Creativity plays a key role in producing and interpreting both mathematics and the arts. Viewers of Mathemalchemy can explore these processes by approaching the work with an open mind and asking questions. Is there a pattern here? Could I find similar patterns elsewhere? What does this diagram mean? What materials did the artists use and why? How did they make this? How could I make this?
“Is there a pattern here?”— Susan Goldstine
What are some good ideas that did not make it into the final installation?
We included as much as we possibly could. As long as someone was willing to make it, I think it went in. There were some proposals for the design of the lighthouse, for example, which did not make it in because we didn’t know how to fabricate it.
Read also: Lighthouse – Fabrication
As a whole team and in sub-teams, we imagined several narratives underlying different scenes, and those narratives evolved over time–variously motivated by mathematical, artistic, and narrative considerations.
One example is the evolution of how Tess the Tortoise would carry her lunch up Integral Hill. Originally she was going to carry it in a backpack. But how would a tortoise put on a backpack? And would the backpack block too much of the tessellation on her shell? Maybe a cart would make more sense, and offer additional surfaces on which to illustrate mathematical concepts–but how would Tess hitch herself to the cart?
After considerable discussion, the team decided that Tess should carry her lunch on a Sierpinski kite, which seemed way more fun–mathematically, artistically, and narrativically–than a backpack or wagon could ever be.
What I particularly like about this example is that I see abundant reasons why the kite was the best choice (among the finite choices we came up with)–but there are other parts of the exhibit where good ideas were excluded simply because we didn’t have enough room to include everything we desired. An example is the Cavalcade of mathematical pages being tossed by the adult silhouette. The gestalt conveys “mathematical jottings on paper,” but viewers who look closely at the mathematical details included therein will observe that we excluded an infinity of good ideas (although in practice, we considered, and excluded, a few dozen additional concepts.)
Read also: Cavalcade – Mathematical Connections
How can mathematics inform literary or musical works of art?
Connections between music and mathematics have interested philosophers and scholars across cultures for thousands of years. One famous example is the Pythagorean concept of musica universalis (also known as the music of the spheres), which theorized that planetary orbits produce musical resonances. Musicians have explored the mathematics of pitch, interval, modes and scales, harmonic progressions, meter, tempo, acoustics, tuning systems, musical forms, motivic symmetries–the list goes on and on. Math has been used not only to analyze music, but also to generate it, from building motives and underlying formal frameworks based on specific mathematical relations (e.g. the Golden ratio) to using set theory and abstract algebra to generate pitch matrices and motives in Western atonal music.
“One famous example is the Pythagorean concept of musica universalis (also known as the music of the spheres), which theorized that planetary orbits produce musical resonances.”— Elizabeth Paley
As for literature, there is an explicit allusion to a math-centered novella in Mathemalchemy itself. If you look carefully among the flowers in the garden, you may spot one or two Flatlanders, the polygonal inhabitants of Edwin Abbott’s two-dimensional world Flatland. Abbott published Flatland, a social satire disguised as a multi-dimensional mathematical treatise (or is it the other way around?) in 1884. The subsequent centuries produced many derivative novels, including Sphereland by Dionys Burger, a direct sequel to Flatland; Planiverse by A.K. Dewdney, in which people from our world contact a being from a two-dimensional planet; and Flatterland by Ian Stewart, which explores the non-Euclidean plane, not unlike our garden.
We could literally go on for pages about all the other mathematics-tinged works of literature, but let us just mention a few more notable works in the English language. Highlights in the realm of theater include Arcadia, a Tom Stoppard play that centers around a young prodigy, her mathematics tutor, and chaos theory, and Proof, David Auburn’s play about a woman struggling with the legacy of her mathematician father and her own mathematical gifts. Clifton Fadiman edited two expansive collections of mathematical stories, poems, illustrations and other curiosities, Fantasia Mathematica and The Mathematical Magpie, that gather works of Isaac Asimov, Samuel Beckett, Arthur C. Clarke, Martin Gardner, Robert Heinlein, Aldous Huxley, Edna St. Vincent Millay, Mark Twain, H.G. Wells, and many many others. Both of Fadiman’s collections also include pieces by perhaps the most famous mathematician/author of recent centuries: Lewis Carroll. Carroll is the pen name of mathematics professor Charles Dodgson, and his timeless classics Alice’s Adventures in Wonderland and Through the Looking-Glass, and What Alice Found There contain countless passing references to and parodies of mathematics and other academic topics.
Explore: Through the Mathemalchemy Looking Glass
Do you identify in any way with any of the characters? If so, who, and why? What is a character you particularly like and why?
Personally, I identify most with Harriet. She likes to collect and exchange objects of mathematical beauty, and I think that is how I approach my mathematical art. As different ideas or craft techniques catch my eye, I see ways to combine them into something tangible that I can share with my community. Also, I don’t have any particular formal training in the arts, so I can sympathize with her artistic impostor syndrome.
“It was great interacting with the other Mathemalchemists — that was the best part.”— Bronna Butler
What’s something new that you learned about mathematics or art as a result of your engagement with this project?
I am very grateful to many talented, creative mathematicians participating in Mathemalchemy who let me “tag along” with them while their visions of math and art were collectively built.
The article mentions (w.r.t. mathematical education) that “Rigor is particularly popular these days, possibly at the expense of intuitive understanding. . .”. How can we start to incorporate intuitive understanding into mathematical education?
Rigor and intuition have been playing tug-of-war with mathematics for centuries. In education, there are already many intuitive tools – drawing a graph of a function, playing with manipulatives, etc. These days there are many other informal resources of intuitive explanations available, for example on YouTube. The challenge in today’s educational climate is that it is difficult to test for intuitive understanding. It is far easier to test for getting the correct answer, and so there is a tendency to teach with that as the only goal.
“Rigor and intuition have been playing tug-of-war with mathematics for centuries.”— Henry Segerman
The article describes: “changing the narrative: now, two inquisitive chipmunks actively engage in collaborative, hands-on play with tangible three-dimensional objects, reveling in their discoveries as much as in the game’s outcomes. We similarly hope Mathemalchemy itself offers viewers a vehicle for joyfully engaging with mathematics.” What other parallels between the vehicle of creative expression in Mathemalchemy and the subjective experience of your audience do you hope to engineer?
From the beginning, we planned Mathemalchemy to be an overabundant piece, with arcane details and secret flourishes tucked away in various corners throughout. We had many reasons for choosing this aesthetic, including our collaborative process and the desire to make return visits to the installation rewarding. But equally important is the viewer experience this produces, a sense of being overwhelmed by everything there is to explore. In mathematics, a good answer leads to a cascade of new questions, and everywhere you look, there are further concepts to investigate.
How would you summarize your Mathemalchemy experience and the goals of the piece in just two or three sentences?
The completed installation, integrating everyone’s contributions together, is exceptionally creative and imaginative. Mathemalchemy’s goal is to connect and to inspire.