# Shortest Paths, Soap Films, and Minimal Surfaces

Michael Dorff during his MAA Carriage House Lecture

“I just happen to have with me today this bucket filled with soap solution, water, and some glycerin,” Michael Dorff told listeners at the start of an MAA Carriage House lecture on October 10.

The Brigham Young University professor and director of BYU’s Center for Undergraduate Research in Mathematics stood in front of a table draped in plastic and crowded with skeletal Zometool creations and deconstructed Slinkies.

“This is a very hands-on presentation. I’m not sure the MAA is used to this this,” he joked.

Used to it or not, MAA was pleased to host Dorff’s talk, entitled “Shortest Paths, Soap Films, and Minimal Surfaces.” Currently spending a sabbatical as a visiting mathematician at the MAA, Dorff is a coauthor of a book* recently published by the organization.

His grander goal was to get the “message across that math is a fantastic thing to learn and to know and that there’s lots of benefits in knowing more mathematics,” and he offered a list of benefits. But Dorff began his lecture with a simple-to-state problem of neighborhood accessibility.

Suppose there are four houses, one at each corner of a unit square. You want to build roads connecting the houses, such that it is possible to drive between any pair. How can you connect the houses using the minimum amount of pavement?

You might connect the houses in a square, in which case the road would be four units long. You could remove one side of the square, thereby reducing the amount of pavement by a quarter. Connect the houses with an X, and you bring the length of the road down to approximately 2.83 units.

To find an even more pavement-conserving solution, though—one that turns out to be the optimal one, though Dorff didn’t take the time to prove it—you can deploy a combination of the Pythagorean theorem and calculus—or a bucket of soapy water.

Dorff dipped into his bucket a pair of acrylic sheets held closely together by screws arranged in a square. When he withdrew the sheets and held them up in the light of the projection screen, the audience could see the screws connected by a path in the shape of an H on its way to becoming an X by being pinched at its midsection, with 120-degree angles at the pinch points.

Dorff demonstrates the shortest path between four points.

Perhaps encouraged by Dorff’s earlier assurance that “it’s not bad to vocalize our appreciation of mathematics,” the audience greeted this soapy revelation of the shortest path with a round of applause.

Dorff next bumped the problem up a dimension, considering not the minimization of distance in the plane but the minimization of area in space.

He dipped an assortment of plastic and metal frames into his bucket, the resulting soap films eliciting oohs and aahs from onlookers.

Watch a slideshow of Dorff's soap films: http://www.flickr.com//photos/maaorg/sets/72157631792112241/show/

A segment of Slinky with a straw joining its ends yielded something reminiscent of a corkscrew.

Two metal rings gave a catenoid, a sort of tunnel indented in the middle.

One frame was a rectangular prism with a pair of opposite edges removed on each short end. Before he dunked it, Dorff said that when he gave a similar lecture at a prestigious university in the Midwest,” a student predicted that the film produced by this frame “was going to look like Venus de Milo.”

In reality, what you get, as Dorff put it, is “pretty much a saddle surface.”

Soap films model minimal surfaces, Dorff explained. Minimal surfaces have a mean curvature of zero at every point, meaning that, at each point, “bending upward in one direction is matched with bending downward in an orthogonal direction.”

The catenoid is a minimal surface, as is the helicoid, that corkscrew formed around the Slinky segment. Enneper’s surface is “like a Pringle’s potato chip,” and Dorff had brought along a finite representation—all minimal surfaces do go off to infinity—of the Costa-Hoffman-Meeks surfaceproduced by a 3D printer.Dorff demonstrated an applet (MinSurfTool) that lets the user examine minimal surfaces from different angles and shows how one may be transformed into another.

The pleasing aesthetic of the transformation that takes the catenoid to the helicoid brought Dorff back to the reasons for learning mathematics he had cited at the beginning of his lecture. He displayed the list again as he concluded his remarks, with “Math is beautiful” appearing this time in red.—Katharine Merow

*Dorff recently wrote Explorations in Complex Analysis with Michael A. Brilleslyper, Jane M. McDougall, James S. Rolf, Lisbeth E. Shaubroek, Richard L. Stankewitz, and Kenneth Stephenson. The book is available in the MAA Store and the MAA eBook Store.