The Bean, officially titled, “Cloud Gate,” is a 33-foot high, 66-foot wide, bean-shaped work of art that sits prominently in Chicago’s Millennium Park. Created by artist Anish Kapoor for the city of Chicago, the 110-ton stainless steel structure is modeled after a drop of mercury. Like a fun-house mirror, it reflects the city and its observers in distorted shapes.
Visitors are often seen exploring and taking photos of their images reflected on the surface of the Bean. But for Wheaton students Gary Babatz ’12, Nathan Bliss ’12, and Nate Veldt ’13, who have studied differential geometry with math professor Dr. Steven Lovett, the giant, shiny legume inspired mathematical questions. Can one reconstruct a specular surface from its reflection? Or can one at least determine its curvature at certain specified points based on the reflection?
Gary, Nathan, and Nate began their summer project by developing instructional materials to support Dr. Lovett’s textbook, Differential Geometry of Curves and Surfaces (co-authored with Dr. Thomas Banchoff of Brown University and published in March 2010). Intended for junior and senior math and physics majors, the book explores what geometric properties of objects become accessible once one has all of calculus and linear algebra at one’s disposal.
In other words, this team has been around the proverbial block with differential geometry. When the time came to select their summer research project, the downtown landscape sparked an idea. “Inspired by the Bean in Chicago, they chose to study what differential geometric properties one may determine from how a curved reflective surface distorts the image of its surroundings,” Dr. Lovett explained.
The team took the train into Chicago, equipped with a camera and two poster boards with square grids drawn on them, and walked to Millennium Park, the home of the Bean. As they began taking pictures, a security guard insisted they obtain a permit from the Cultural Center of Chicago to take pictures of the structure. After receiving official permission, they began taking pictures with the grid in the reflection of the Bean.
“We enjoyed many mathematical conversations with tourists who were interested in our experiment,” said Dr. Lovett.
After experimenting with various curved surfaces, flat mirrors, and the Bean itself, they began to try and reconstruct the shape of the Bean using formulas they developed for projective transformations and data gathered from pictures. They soon realized how difficult this problem actually was.
“We attempted a variety of techniques to reconstruct the specular surface, including polynomial best fit models and triangulated surfaces. Though our specific numbers have suffered from experimental error, we can distinguish from a picture whether a surface has positive, zero, or negative Gaussian curvature. This is a geometric property that helps quantify the curviness of the surface,” Dr. Lovett stated.
For the last two weeks of the project, the team travelled to Brunswick, Maine, and worked on the campus of Bowdoin College. They were sponsored by Dr. William Barker, chair of the Bowdoin College Math Department. “We lived communally and enjoyed the tastes and sights of Maine during the evenings, including exploring various forms of lobster: twin lobster meals, lobster mac & cheese, lobster chowder, lobster rolls, and even lobster ice cream.”
For these three math majors and their professor it was a unique learning journey, applying mathematical principles to a Chicago landmark. For Gary, Nathan, and Nate, the project exposed them to what research in advanced or applied mathematics could look like and helped reaffirm their future plans to either study or apply mathematics in graduate school.
Nate expressed his interest in graduate school in mathematics, noting the verse, “Whatever you do, work at it with all your heart, as working for the Lord, not for human masters.” (Colossians 3:23).