Bubbles and Levity for Those Who Are Calculus Inclined

Michael Dorff, the Director for the Center of Undergraduate Research in Mathematics and a professor of Mathematics at Brigham Young University, joined St. Mary’s students in a discussion of the minimal surface properties of soap bubbles during the Natural Science and Mathematics Colloquium lecture “Mathematics, shortest paths, soap bubbles, and the shape of the universe” on Feb. 16.

Dorff began the lecture with a mathematics problem, drawing four dots on a chalk board in the Schaefer Hall lecture room that formed the four corners of a square.

He asked the audience what the shortest path would be between all of the dots that would connect all of the dots together.

After giving the audience some time to work on the problem, Dorff listened to audience suggestions before displaying the correct answer on the screen: each pair of opposite-side dots forming the base of a triangle of small height, with the tips of the triangle linked by a straight line, certainly not as easy as connecting all dots together in the shape of a box.

Dorff showed a way to calculate the length of this shape given that adjacent dots were one unit away from each other before approaching the main points of his talk.

“Mathematics is the art of asking good questions,” Dorff said.  “When people think of the math behind calculus, they think of the derivatives and integrals.”

In reality, according to Dorff, while this is true, calculus also relates dimensions of objects, such as the first dimension (distance and time), second dimension (speed and area), and third dimension (acceleration and volume), all reached by taking the derivative of the preceding dimension or through integration.

Soap films, or the shape of soap bubbles between or on objects, represent model surfaces of minimal surface area, similar to how the shape between the four dots in the first example during the lecture represented the smallest distance between all of the points.

“You have this surface tension,” said Dorff, explaining the phenomenon, “that minimizes the surface area of the shape.”

Dorff brought several ball-and-stick models of shapes and a large soap bucket to demonstrate this point, showing that soap bubbles will form minimal-area surfaces between points of contact.

Soap bubble shapes depended on the shape used as the framework, and ranged from ennepers, catenoids, and scherks to periodic surfaces, double ennepers, and twisted scherks, all names for unique three-dimensional shapes designed to decrease surface area of the soap bubble inside.

Dorff applied this idea to another minimal-surface example, known as Melzak’s problem, by asking what the smallest volume would be for a surface of unknown edges of length equal to one unit (that is, how many edges would be needed to create the smallest volume).

The answer was a tetrahedron, of volume 0.118 cubic units in that case.

Changing the problem, Dorff asked what the smallest sum of all edges would be to reach a fixed volume of one cubic unit.

While a cube would have 12 units of edges, and a tetrahedron 12.238 units, the smallest edge length is still unknown.

Dorff approached several of the problems in terms of dimensions, always going an extra dimension higher to analyze a problem.

He applied this idea to his final example: the minimal area of the universe.

He explained that the Earth was once thought to be flat (two-dimensional), and that if one sailed off of the Earth, he would never return.

But, we now know the Earth to be spherical (three-dimensional), and if we travel all the way around the world, we return in the same position we started from the other side.

In the case of the universe, if we thought it was two-dimensional, then one could fly in a straight direction and never return.

If it were three-dimensional, one would travel through the universe and arrive at the same place, only from the opposite direction.

While the answer is still unknown, mathematics is one field that could attempt to explain it.

Dorff’s lecture to students, faculty, and staff seemed to the audience to be understandable and interactive, and participation was  encouraged by Dorff’s distribution of prizes to those who correctly answered questions throughout his lecture.

“I like soap bubbles, a lot,” said Josh Kaminsky, a sophomore philosophy major who attended the lecture, and won one of Dorff’s prizes for answering the first example question correctly.

“Minimizing things is something I’ve been interested in since geometry, so this talk was the college version of that.”