Curious Area and Volume Ratios – A Discovery

Mar 23, 2018 | Stacey Ortman, director of public information

Dick Smith, associate professor of mathematics at the University of Dubuque, and Kris Nelson (C’15) worked together on “Curious Area and Volume Ratios.” Nelson helped create a physical model of the math problem that was started by Smith.

DUBUQUE, Iowa – An unlikely friendship that led to a curious discovery started in a math classroom in the University Science Center on University of Dubuque’s campus.

Kris Nelson (C’15) enrolled in classes at UD after serving active duty in the Army and working in the private sector. His goal was to obtain an undergraduate degree in computer information technologies. Even though he enjoyed math, it had been 20 years since he had studied numbers and he was nervous.

Dick Smith, associate professor of mathematics, saw potential in the student veteran who challenged him to rethink equations. Smith encouraged Nelson to consider becoming a mathematics major.

“He wasn’t just a consumer of what was delivered. He also challenged me,” Smith said. “He would come up to me after class and very politely say, ‘I’ve had an idea.’”

One of those ideas occurred after Smith shared a problem, “Curious Area and Volume Ratios,” with his modern geometry class. He had been working on the problem since 2010, some of that time with Dan Flath, PhD, a former visiting professor at Macalester College who is now an independent mathematical consultant with Flath Math Results. Smith shared how when he looked out the window of an airplane and saw an irrigation field, he wondered what would happen if the field increased in size but the end of the irrigation tubes remained fixed. Would the ratio of the whole thing become predictable?

After class, Nelson asked Smith if he had considered moving the problem into three dimensions.

“Two old professors were stuck and this guy who had been away from math just blows us completely out of the water” Smith said, laughing.

Nelson combined his computer information technology knowledge with his math knowledge to extend “Curious Area and Volume Ratios” into three dimensions. Mark Woodhouse, assistant professor of computer information systems at UD, used Nelson’s dimensions to create a physical model of the problem.

“It was an interesting idea that started out in two dimensions and I wondered what it would be like in three dimensions. I also wondered if the relationships that held in two dimensions would also hold in three dimensions,” Nelson said. “What I discovered was the same formulas held true in three dimensions, with the exception of it being to the third power versus the square.”

In 2016, Nelson presented “Curious Area and Volume Ratios” at the Minnesota Regional Mathematical Association of America conference at the University of Minnesota. The problem was created via an octagon by drawing two line segments from each vertex of a square to the points on the square one unit from the opposite corner. The ratio of the area of the square to the octagon produces triangular numbers. As the square gets very large, the octagon approximates a square with an area ½. A 3-D extension draws three segments from each vertex of a cube to points one unit from the opposite vertex, producing a solid with 24 edges. When the edge of the cube is an integer, the ratio of the volume of the cube to the solid is always a whole number. As the cube gets very large, the solid’s volume approximates ¼.

“It’s absolutely beautiful. Everything works and the ratios are definitely curious,” Smith said.

Nelson added, “The most interesting part about it is we need one piece of information to determine everything about the solid in the center. As long as we have that one number, we can determine everything.”

Nelson, who was a triple major in computer information technologies, mathematics, and business, earned a bachelor’s degree in December 2015. He will graduate from the UD Master of Business Administration program in May.

Smith continues to work on the problem.

“This is what mathematicians do,” he said. “This problem was there and we decided to wonder up this mountain and try to find what we could find up there.”