Under ordinary circumstances, “Lissajous” might be the name of a French actress or a densely caloric patisserie dessert. But this is engineering, so the reality is a bit more complicated.
Lissajous are a way to look at the relationship between two wave frequencies whose differential has been cast into form – plotted out into curving, intersecting lines. They can describe motion, which is exactly what intrigued College of Engineering students Alexandria Will-Cole and Isabella Mendoza so much. They completed a project based on Dr. Leslie Lamberson’s National Science Foundation CAREER award investigating movement. The project inspired a dance company, drove the choreography and music, and coaxed the concept of “Pi” out of its figurative hiding place. All at once.
The dance piece resulting from their collaboration, “Lissajous,” will be performed at the John F. Kennedy Center for the Performing Arts in Washington, DC this May by the Bowen McCauley Dance Company, with Will-Cole and Mendoza planning to sit pridefully in the audience.
How a dance project themed around the irrational number Pi and Lissajous lines came to be a public performance is really part of the larger story of how science intersects with artistic expression in unexpected, vastly relatable ways.
“As we get more and more interdisciplinary, we’re going to be looking at things that are seemingly unrelated where you can really make some interesting discoveries,” said Will-Cole. “We wanted to do an outreach project that could inform the public about Pi. Pi relates to dance through symmetry, rotational symmetry, and rotational motion. But it also relates to frequency. So we thought we could look at frequency, and that led us to this idea of the Lissajous figures. I remember plotting them in physics coursework in my undergraduate days.”
Using sensors called accelerometers that were attached to members of the dance company, Will-Cole and Mendoza collected data on the rate of acceleration of dancers’ movements. When these were plotted against the acceleration rates of other dancers, the resulting Lissajous lines revealed patterns that were then used by the company’s director to choreograph new dance steps and movement phrases.
“Lissajous were these beautiful figures you could make based on looking at the frequency and amplitude of two different waves on two different axes,” Will-Cole added. “Depending on the relative amplitude of the waves or the phase between them and how that relates to Pi, they can give you different shapes – it could give you a circle, it could give you a line, it could give you this really complex beautiful butterfly-like patterns.”
Will-Cole is a third-year PhD candidate in the Department of Materials Science and Engineering (MSE). Mendoza is graduating with a BS in MSE this spring and plans to attend graduate school. Both are former dancers; Will-Cole at programs such as the Peabody Institute and Central Pennsylvania Youth Ballet. Mendoza danced throughout her childhood and continues taking dance courses here at Drexel.
Lamberson’s Former Company
The advisor on the project was Dr. Leslie Lamberson, an associate professor in the Department of Mechanical Engineering and Mechanics (MEM). It was Lamberson who drew the students together, introducing them to the company she herself had danced with professionally for two years while a postdoctoral research scholar at Johns Hopkins University. Bowen McCauley is an award-winning contemporary dance company based in Arlington, VA that is celebrated for the athleticism of its dancers.
“Lissajous” was premiered to sold out crowds in Washington, DC at the Dance Place venue last December, coupled with music composed by a musician from the University of Florida. Individuals in the audience – everyone from professors to artists to “regular” theater-goers – approached Will-Cole and Mendoza afterwards, impressed with how seamlessly the worlds of engineering and dance had been united. The piece was commissioned by a grant to Lamberson from the National Science Foundation, and sponsored by Drexel University.
“The goal of this project was to utilize a ‘scientific lens’ to work together by quantifying and hence inspiring the creation of performing arts – in this case, music and dance,” said Lamberson. “I find it so interesting that we can use scientific tools and measurement techniques to explore creative spaces. It seems paradoxical or counterintuitive, but it’s not.
“Isabella and Alexandria far exceeded anything I could have imagined,” she added. “They came up with all the nuanced details of the measurements, and even with the awesome idea of the Lissajous representation for visualization. I could not have done this project without them.”
When Will-Cole and Mendoza first met with the company dancers, they gave them a primer on how Pi relates to dance through the mechanics of motion -- how dancers rotate, for example, or how they counterbalance based on symmetry. The Bowen McCauley dancers were quite receptive to the scientific input, partly because they already have an intuitive sense of physics and mechanics and how they tie into musical chords that have to be coordinated with dance steps.
“We told them how you literally see Pi in a circle: you take the ratio of the circumference of the radius, and that’s Pi,” said Mendoza. “But it also relates to frequency and waves, and a lot of people don’t think about that. So, we were just trying to encourage this deeper curiosity into what’s out there. Artists and scientists are often butting heads. But hey, both our ways kind of connect.
“The dancers were very excited,” Mendoza added. “That’s what made the data collection less stressful. They were really into the accelerometers and they kept asking, ‘What if we try it on this leg or with this motion?’ ”
Under the guidance of Lamberson, Will-Cole and Mendoza have plans to extend the outreach component of the dance project, taking it to local Philadelphia middle and high schools to encourage a rapport with the STEM field. Young girls and boys may not see an immediate connection between pursuits that appeal to them and the field of science, said Mendoza. Through “Lissajous,” they can enjoy a demonstration of how science doesn’t have to be segregated into a seemingly inaccessible domain.
“You know, these possibilities are everywhere,” Mendoza explained. “Even if I’m walking down the street with my friend – if I’m walking at a certain pace and my friend’s footsteps are just a little bit different, I could take that and plot it, and something would happen. It all relates to Pi and symmetry.”