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# Math Dance: Bringing the Process to Life

## Person, Product, But Where’s the Process?

At first glance, math and dance seem an unlikely pairing for interdisciplinary projects. While mathematics is highly creative, it is not typically thought of as performance. Equations often appear static. Many math papers are written as a snapshot of the finished product and rarely show the research process or timeline of the project. The mathematics community is working hard to promote the lives and stories of the people who do mathematics. Where we continue to fall short is promoting the process of doing mathematics.

After reading a clever proof, I often think how did the author come up with this amazing idea? Many times the author has an interesting visualization or analogy in their mind that allows them to see a larger picture. However, these imaginings are rarely included in the most elegant proofs put to publication. The mathematical characters, stories, narratives of life and death, analogies of love and hate, are in the mind of the mathematician, but hidden to the reader.

For example, the system of differential equations

has an oscillating solution

While mathematically complete, we can more richly understand this system of equations with a narrative. Strogatz explains how we can see this system of differential equations as the story of a love affair between two people

While there likely aren’t as fitting analogies for all mathematical problems as there is for the love affair differential equations, looser analogies can still give great insight into solving a problem. Passing through the halls, I have heard casual conversations between mathematicians where they refer to elements of a set as people, and functions as moving people from one area to another. These causal analogies may not be rigorous enough to include in a publication, but they bring life and humanity to the mathematics. These analogies need to be told! I personally use dance as an exceptional communication tool to bring mathematics, and my process of doing mathematics, to life.

## Why Dance?

Many of the metaphors and imageries used in the process of doing mathematics are about people and their movements or transformations. Dance, as a metaphor, embodies all aspects of human behavior in an elegant and visually exciting manner. It is a fabulous tool for elaborating, clarifying, and explaining difficult mathematical concepts to the nonexpert.

For someone who already wants to learn mathematics, there are plenty of educational resources available. The goal of a Math-Dance is not limited to teaching mathematics, but to share a moment of joy and the beauty of mathematics with someone who doesn’t already value mathematics.

An important step for successful communication is to determine who is your target audience. In showing the process of doing mathematics, the goal is to improve mathematics’s public relations. We need to fight widespread math anxiety and empower people who consider themselves “bad at math.” To reach this audience, we need to use culturally relevant avenues of communication; dance is one of many. While I have performed several live Math-Dance pieces, I prefer to make videos as they will reach a much wider audience than a single live performance.

I am not alone in my use of dance to promote math and science. *STEM From Dance* is an outreach program dedicated to uplifting young women of color by using dance to promote careers in STEM. One of *STEM From Dance*’s guiding principles is the cultural relevance of dance in communities of people of color.

Our communities have traditionally used dance to release, celebrate, draw strength, communicate, and learn. Combining dance with tech transforms STEM careers from unthinkable to within reach.

*Science Magazine* annually runs an international *Dance Your PhD* competition challenging graduate students to communicate their research in dance videos. John Bohannon, the founder of this competition, gave a TED talk where he describes his motivation for using dance to communicate science.

I have a PhD in Molecular Biology. I still barely understand what most scientists are talking about. So, as my friend was trying to explain [an] experiment [to me], it seemed like the more he said, the less I understood. Because, if you’re trying to give someone the big picture of a complex idea, to really capture its essence, the fewer words you use, the better. In fact the ideal may be to use no words at all. I remember thinking “My friend could have explained that entire experiment with a dance.”

I won the Dance Your PhD competition in 2017 with a Math-Dance video I made about *Representations of the Braid Groups* *Fist and Heel Performance Group* to to find mathematical structures in African diasporic movements and performance traditions

## Bringing the Process to Life

It is not the daily routine of doing math research that I want to showcase, but rather the imaginative mathematical thinking that happens when doing research. How does a mathematician think about math? See math? What does it feel like to do mathematics? These are the processes I strive to share through my Math-Dance videos.

Each theorem I prove has a soundscape, landscape and movement. Linear algebra feels different than topology, analysis feels different than algebra. I want to express this difference to the general public without having to first teach them five years of prerequisite mathematics.

In 2019, I made the Math-Dance video titled “Algebra, Geometry, and Topology: What’s the difference?” This video is publicly available on YouTube *We Are Mathematics*. I wanted to showcase how the fields of algebra, geometry, and topology are different. I used three main communication techniques to achieve this goal. First, I made an analogy with something that the general public is already comfortable with. I compared the different fields of mathematics to the different fields of science—biology, chemistry, physics, as shown in Figure 2. I explained that what makes the fields of study different is not that they study different objects, but they use different techniques to study the same objects.

The second communication technique I used was to simplify the message as much as possible. To show how algebra, geometry, and topology are different, the video showed how each field might study a circle, and only a circle. The video did not discuss other shapes and kept the scope very small.

The third technique employed in this video is the use of soundscape and landscape. Each mathematical field used a different colored lighting scheme, different types of circular props and had their own musical melody. Geometry danced with a rigid circular hula hoop, topology with a flexible loop of fabric, and algebra was a dance of laser lights, as shown in Figure 3. The audience can see, feel, and hear a difference between the fields of mathematics, without requiring a technical understanding of the fields.

I commissioned the score for this video from accomplished musician James Whetzel. Partner of a mathematician, Whetzel was primed for our intriguing discussion of how the different fields of mathematics could sound. Geometry is made of rigid flows with a repetitive cyclical nature. Topology is quirky and throws away the rules and structure of geometry. Algebra is discrete and electronic with staccato punctuation. Whetzel tangified these abstract descriptions into the beautiful score titled “This is What Topology Sounds Like.”

For many, the best part of mathematics is understanding why a result is true, or why a proof works. In 2023, I made a Math-Dance video that shared an “ah ha” moment with the viewer. The video titled *Alexander’s Theorem* explains the visually straightforward algorithmic proof of Alexander’s Theorem. This video is publicly viewable on YouTube

Alexander’s theorem is a foundational theorem in knot theory which states that every knot is isotopic to the closure of a braid. Since knots are physical objects in our 3D reality, I did not need to use a carefully chosen analogy to describe the theorem. Instead, the key to performing Alexander’s theorem was to find the right language to describe braiding. In the video, instead of saying “closure of a braid,” I described when a knot is “braided around an axis.” The axis was represented as a pole in the dance studio. Using this definition, it was natural to describe a braided knot using acrobatic pole dancing and clockwise versus counterclockwise motion around the pole. To contribute to the visual clues, I used different colored ambient light that changed color with clockwise versus counterclockwise motion, and the knot itself lit up in different colors to intensify the direction changes.

Before Alexander’s theorem was stated, the video compares when a knot is and is not braided around an axis. There are two longer dance scenes, one dance of a braided knot and one of a not-braided knot. I wanted the viewer to see beauty and feel fondness for the braided knots, and feel dislike or ill-at-ease for the not-braided knots. That way, when the theorem was stated, the viewer would emotionally appreciate being able to turn a not-braided knot into a braided one, even if the viewer could not mathematically appreciate the value of the theorem. I used musical, lighting, and dance techniques to achieve these emotional responses.

Figure 4 shows two video stills from *Alexander’s Theorem* during the braided knot dance sequence and the not-braided dance. To encourage feelings of harmony and fondness during the braided dance, there is soft lighting illuminating only the dancer so she appears to float against the dark background. The music swirls in waves with a high-pitched enchanting melody. The knot illuminates with a continuous rainbow that flows around the pole as the dancer moves. The dance style is graceful, balletic, and fairy-like. The dancer freely spins with a clockwise rotation for the entire dance.

Comparatively, during the not-braided dance, the dancer is backlit with colored light that chaotically changes between blue and red. The atmosphere is thick with haze and the room appears musty or blurred. The music has deep rumbling bass notes with stark percussion. As the dancer attempts to spin around the pole, a second person dressed all in black reaches out, harshly prevents the dancer’s motion, and throws her in the opposite direction. The dance style is exasperated, exhausted, and chaotic with many direction changes. During this scene, the dancer is using an aerial pole strap attached to the pole. With the strap, she can hang and spin at the same time which amplifies the directional changes.

While these two dance sequences were not mathematically required to describe the theorem, they provide something that published mathematics cannot do well; they invite moments of emotional connections for the nonexpert viewer. The dances give insight into the imaginative way I think about braids as a braid group theorist. These dances bring my research process to life.

## Math-Dance FAQ’s

I am frequently asked three questions about my Math-Dance projects:

- (1)
How do I come up with a Math-Dance video concept?

- (2)
How has making Math-Dance projects affected my process to create new mathematics?

- (3)
How long does it take to produce a Math-Dance video?

To answer the first question, it takes practice and permission to create a Math-Dance. If you think it’s impossible to dance your math, then it is impossible. If you give yourself permission to imagine it is possible, you will be amazed at the ideas you come up with.

I make it part of my research routine to ask myself if I could dance this math. What I’m really asking is can I describe this concept by telling a story about people in plain English (with no technical terms). Then I ask myself if I can tell the story with movement and props. It is a two-part translation: math to English, English to movement. As an avid dancer, I am constantly learning new styles and new aerial apparatuses. I incorporate Math-Dance into my dance routines as well. I ask myself what kind of math can I describe with this dance? This is how I conceived the story for *Alexander’s Theorem*. For other examples of how I turned math into dance, see

To elaborate more on the second question, while doing research, asking if I can dance my math helps me to creatively experiment with ways to visualize the problem and see it from a different perspective. This practice has improved my communication abilities, but I have not noticed my mathematical understanding to be greatly affected, yet. My main goal for Math-Dance is to communicate ideas to others. My creative process is to strip down the mathematical ideas to a simple streamlined narrative. In this streamlining, I am looking for emotional and visual representations of the mathematical qualities, but I lose the mathematical rigor that I need in order to prove results. The major benefit I have received from sharing my Math-Dance projects has been the ability to find like-minded collaborators in a much wider network.

To answer the third question, I will describe the timeline and production process of my most recent project, *Alexander’s Theorem*.

I had the idea for a pole-dancing video about Alexander’s theorem in the summer of 2022. I was visiting collaborators to work on a project that we jokingly called our “pole dancer project.” We took two pole dancing classes together as a social activity during the research visit, and this gave me the idea to make an Alexander’s theorem video using pole dancing. I secured funding from Elon University for the project in September of 2022 and I wrote a storyboard for the piece, which can be viewed on my website

Through April and May, I decided on the programmable LED lights to represent the knot. There were many different ideas of what material to make the knot out of. Ultimately, I decided on the LED lights because the orientation flow of the knot was pivotal to the algorithm and I needed the knot to show the flow. These lights had to be meticulously programmed and run through an Arduino. Every time the rope was moved or the colors needed to change, I had to pause filming, write a computer program on my computer, and reload it to the Arduino. It was painstaking, but so worth the effort.

The video was filmed in my garage over two weeks in early July, 2023. I hung black paper over the walls and black foam flooring to cover the concrete. Some behind the scenes photos can be seen in Figure 5. I recruited three undergraduate students to help with the filming process: two students to operate the cameras and one to be my assistant. I borrowed all of the filming and lighting equipment from the media services center at Elon. There was a total of 11 hours of filming over two weeks. We could only film late at night as it was very hot and humid in my noninsulated garage in the North Carolina summertime. Under usual circumstances, the filming could have been done in one weekend, as were my other projects, but it took two long weeks for this project. The heat also led to unforeseen delays as effective pole dancing requires dry skin contact with the pole for friction to hold your body off the ground. I had to simplify much of the choreography to account for the inability to stay off the ground for more than a few seconds.

After the filming was complete, I edited the footage in about one week and posted the final cut to YouTube at the end of July 2023. This was my first experience filming and editing. I used Adobe Premiere Pro to edit the footage. I learned to use this software by watching online tutorials over the course of two weeks. For my previous videos, I hired a professional, Alex Nye, to film and edit the footage.

## References

[ Boh] - John Bohannon,
*Dance your PhD—John Bohannon and Black Label Movement, TEDxBrussels*, https://youtu.be/UlDWRZ7IYqw?si=jYCkCtOhf6ZwY8qe.,## Show rawAMSref

`\bib{Ted}{misc}{ author={Bohannon, John}, title={{D}ance your {P}h{D}---{J}ohn {B}ohannon and {B}lack {L}abel {M}ovement, \textup {{T}{E}{D}x{B}russels}}, url={https://youtu.be/UlDWRZ7IYqw?si=jYCkCtOhf6ZwY8qe}, }`

[ Schaf] *Karl Schaffer*, http://mathdance.org/index.html.,## Show rawAMSref

`\bib{Schaf}{misc}{ label={Schaf}, title={\textup {Karl Schaffer}}, url={http://mathdance.org/index.html}, }`

[ Sch17] - Nancy Scherich,
*Representations of the Braid Groups YouTube video*, 2017, https://youtu.be/MASNukczu5A?si=8ITyTcqaLTdtXZaF.,## Show rawAMSref

`\bib{Rep}{misc}{ author={Scherich, Nancy}, title={{R}epresentations of the {B}raid {G}roups {Y}ou{T}ube video}, url={https://youtu.be/MASNukczu5A?si=8ITyTcqaLTdtXZaF}, date={2017}, }`

[ Sch18] - Nancy Scherich,
*Turning math into dance: Lessons from dancing my phd*, Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture (2018), 351–354.,## Show rawAMSref

`\bib{Sch}{article}{ author={Scherich, Nancy}, title={Turning math into dance: Lessons from dancing my phd}, date={2018}, journal={Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture}, pages={351\ndash 354}, }`

[ Sch19] - Nancy Scherich,
*Algebra, Geometry, and Topology: What’s the difference YouTube video*, 2019, https://youtu.be/xgKc7dFz-ko?si=DA-iNZSqjN2qVLlW.,## Show rawAMSref

`\bib{AGTd}{misc}{ author={Scherich, Nancy}, title={{A}lgebra, {G}eometry, and {T}opology: {W}hat's the difference {Y}ou{T}ube video}, url={https://youtu.be/xgKc7dFz-ko?si=DA-iNZSqjN2qVLlW}, date={2019}, }`

[ Sch22] - Nancy Scherich,
*Story board for Alexander’s Theorem*, 2022, https://nancyscherich.com/story-board-for-math-dance-alexanders-theorem/.,## Show rawAMSref

`\bib{Storyb}{misc}{ author={Scherich, Nancy}, title={Story board for {A}lexander's {T}heorem}, url={https://nancyscherich.com/story-board-for-math-dance-alexanders-theorem/}, date={2022}, }`

[ Sch23] - Nancy Scherich,
*Alexander’s Theorem YouTube video*, 2023, https://youtu.be/zhKp6GTVy_4?si=RzulvlxN33Xy7ySh.,## Show rawAMSref

`\bib{AT}{misc}{ author={Scherich, Nancy}, title={{A}lexander's {T}heorem {Y}ou{T}ube video}, url={https://youtu.be/zhKp6GTVy_4?si=RzulvlxN33Xy7ySh}, date={2023}, }`

[ SFD] *STEM From Dance*, https://stemfromdance.org.,## Show rawAMSref

`\bib{SFD}{misc}{ label={SFD}, title={\textup {STEM From Dance}}, url={https://stemfromdance.org}, }`

[ Str88] - Steven H. Strogatz,
*Love Affairs and Differential Equations*, Math. Mag.**61**(1988), no. 1, 35. MR1572691,## Show rawAMSref

`\bib{STRO}{article}{ author={Strogatz, Steven~H.}, title={Love {A}ffairs and {D}ifferential {E}quations}, date={1988}, issn={0025-570X,1930-0980}, journal={Math. Mag.}, volume={61}, number={1}, pages={35}, url={https://www.jstor.org/stable/2690328?origin=pubexport}, review={\MR {1572691}}, }`

[ WW21] - Reggie Wilson and Jesse Wolfson,
*Mathematics and Dance: Notes from an Emerging Interaction*, Notices of the AMS**68**(2021), no. 11, 1926–1929.,## Show rawAMSref

`\bib{Wol2}{article}{ author={Wilson, Reggie}, author={Wolfson, Jesse}, title={Mathematics and {D}ance: {N}otes from an {E}merging {I}nteraction}, date={2021}, journal={Notices of the AMS}, volume={68}, number={11}, pages={1926\ndash 1929}, }`

[ Wol] - Jesse Wolfson,
*https://jpwolfson.com/mathematics-and-dance/*.,## Show rawAMSref

`\bib{Wol}{misc}{ author={Wolfson, Jesse}, title={\url {https://jpwolfson.com/mathematics-and-dance/}}, }`

## Credits

All figures are courtesy of the author.