Matt Metcalf of Homer is spending his summer doing math and hoping for a “Eureka!” moment.

His goal is to generate original fractal tilings, the study of which may someday be used by video game designers, medical researchers or computer encryption specialists.

Metcalf has been drawn to numbers as long as he remembers. A senior dual mathematics and physics major at SUNY Cortland, Metcalf is passionate about the logical thinking needed to learn and do mathematics.

Mathematics is exact. Two plus two will always equal four in elementary mathematics. In more complicated applications, answers are not as obvious, but the same logic applies.

Metcalf is doing just that this summer as one of 10 Summer Research fellows on campus. He is working with Isa Jubran, associate professor of mathematics, as he studies the theoretical foundation of fractal tilings.

A fractal is a complex geometric object whose dimension is fractional. A line drawn on a piece of paper is one-dimensional, a square is two-dimensional and a solid cube is three-dimensional. Fractal objects fall somewhere in between. Some fractals are self-similar, meaning that they consist of parts that have the same shape as the whole fractal. Examples of self-repeating fractals found in nature include trees, lightning bolts and snowflakes.

Metcalf and Jubran are investigating fractal tilings of the plane this summer. A tiling of the plane is a countable family of tiles without gaps and without any overlapping of the shapes. For example, if one were to use squares to tile a plane, it would look like a checkerboard. Rather than squares, Metcalf and Jubran are using tiles that have a fractal boundary to cover the plane.

There are countless real-life applications of fractals:

**–** Video game developers can create massive landscapes by writing code that repeats fractal objects.

**–** The encryption field is using the complexity of fractals to help protect the transmission of sensitive data.

**–** Medical professionals are using fractals to look at repeating patterns in blood vessels in the eyes and lungs to detect or predict disease.

Metcalf and Jubran first conducted a literature review to determine where to focus their research. They investigated a half-dozen articles on fractal tilings. Software such as IFSKit and Wolfram Mathematica is used to create fractal tilings on what looks like digital graph paper.

“We wanted to start out by reading through a bunch of articles and we were hoping to run into open questions along the way,” Metcalf said. “One of the main goals of this project is to generate dozens of original fractal tilings using the theory that we learned by reading these articles. We’re starting to run into a few questions that we may pursue.”

Software allows Metcalf and Jubran to generate fractal tilings. Much of the math involved is discussed in a sophomore- or junior-level linear course such as linear algebra.

Metcalf has spent many late-night hours this summer rewriting code for Mathematica, as some of the literature on how to generate fractals with it used an earlier version. Those revisions will be part of a preprint that Metcalf is preparing to document his research.

“I had to read through all of the fractal tiling code and figure out what was wrong with it because it wasn’t working at first,” Metcalf said. “It took me a lot of time to learn the language and understand all of the fractal tiling code. It would take an hour to understand a certain passage and figure out what was going on.”

Since conquering the software, Metcalf has run hundreds of experiments that have led to discovering patterns and formulating hypotheses and conjectures. The next step is to find out how and why certain fractals he generated behave the way they do.

Metcalf spoke on the earliest portions of his research at SUNY Cortland’s 2017 Transformations: A Student Research and Creativity Conference. He will return to Transformations in 2018 to update his findings and attend other undergraduate research conferences.

Metcalf used some of his stipend to buy a personal license for the Mathematica software so he can work from any location. For students such as Metcalf, a summer research fellowship may mold the rest of their academic career and help them connect with researchers around the world.

“I’m hoping that this will lead to some collaboration between Matt and other researchers who are actively working in the field of fractal tilings,” Jubran said. “They are more knowledgeable in some aspects of the field than I am and will be able to advance the work Matt and I are currently doing. We will continue this project in the upcoming academic year paying special attention towards possibly generating a scholarly article to be submitted for publication.

“More importantly, the research skills he is learning this summer will prepare him very well for life in graduate school,” Jubran said.

Metcalf admits that he hasn’t yet had his big breakthrough. But he continues to dig.

“All of it is new to us, so there are a lot of minor ‘aha!’ moments, such as when you finally understand a concept,” Metcalf said.

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*William Montgomery is assistant director of communications for SUNY Cortland*.