In Denver, Colorado, SU alumnus Robert Lehr and Professor and Chair of Mathematics Fumiko Futamura were presented with the Carl B. Allendoerfer Awards at MathFest, the summer meeting of the Mathematical Association of America (MAA) and one of the largest math conferences in North America. Lehr and Futamura were recognized for their exemplary writing in the field of mathematics in their article “A New Perspective on Finding the Viewpoint,” published in the fall 2017 issue of Mathematics Magazine. “These prize-winning authors serve to inform and inspire our community. We all benefit from clear and effective writing that brings important results to the broadest possible audience,” said Michael Pearson, executive director of the MAA.

Lehr also won an outstanding presentation award from the math honor society Pi Mu Epsilon (PME) for his talk “Perspective Drawing: How to Find the Immersion Point,” which was based on the prize-winning article.

The publication reflects Futamura and Lehr’s interests in applying mathematical principles to other disciplines: Futamura is also an artist who works in oil paint, charcoal, and crochet while Lehr’s flair for creativity and innovation includes illustration and developing brainteasers. In the essay, the authors argue for a new answer to the question of where individuals should stand in front of an image in two-point perspective—that is, an image that has two vanishing points, such as Hendrick van Vliet’s painting Interior of the Oude Kerk, Delft (1660) or Gustave Caillebotte’s Paris Street; Rainy Day (1877)—to view it most advantageously. It’s an article that could not only have impacts on math but also inform theories and practices in art, museum curation, and the psychology of human perception.

“Braindrizzling,” Innovation, and Collaboration

According to Futamura, the essay began as an idea Lehr had while working on a project for her geometry class. She remembers him studying fractals and cross ratios while drawing beautiful pictures on the board, which he later transformed into “gorgeous, intricate” computer illustrations. In those drawings was the germ of their paper. Lehr recalls having that eureka moment while learning about various algebraic and geometric methods of finding the optimal viewpoint of a perspective image. “I was excited at first when I saw how elegant [the geometric approach] looked, but it was impractical,” Lehr says. He wanted to find a better method.

So he sketched and brainstormed—or “braindrizzled,” as Lehr likes to say—during and between classes. Creativity and persistence in solving puzzles are in Lehr’s DNA, it seems: a mathematics major, he’s part of what he calls a “quantitative family,” one that includes mathematicians and software engineers. “I get carried away with ideas,” Lehr admits sheepishly. Futamura shines a more positive light on the SU alum’s proclivities for exploratory thought and discovery: “He’s using his math, he’s using his art. He’s such an inventor type. He’s constantly thinking about taking an existing method and trying to improve upon it. And that’s exactly what he did in this paper.” After devising a new approach to the viewpoint problem, he consulted Futamura about his thinking. “She added incredible depth and knowledge of the field” to his hypothesis, he says, and over the next few months, they collaborated on 13–14 drafts of the article. It was eventually published in a double-blind, peer-reviewed journal that has only a 13% acceptance rate of submitted papers.

Both authors agree that their collaboration was a meeting of minds: While his approach is intuitive, she could supply the rigor necessary for a scholarly publication. But Futamura made Lehr commit his thoughts to paper by writing the first draft before she added her contributions. So what was it like being an undergraduate coauthoring a paper with a professor? “Slow, but rich and collaborative,” Lehr says. “I was happy she gave me a lot of support to finish it because it’s hard for me to be that patient to finish something. I had to be even more patient after submitting for publication: we had to wait two years!” Even then, the journal required revisions, but those stylistic edits were minimal—a testament to the strength of their article’s content.

Overcoming the Odds

When Lehr and Futamura were notified that they’d be receiving the Allendoerfer Award this past April, he was excited to attend the MAA conference in Denver, and he decided to revisit the article and create the prize-winning presentation that he would deliver at the meeting. However, Lehr’s mother anticipated the months-long process of crafting his presentation and the trip itself with some trepidation, with reason: Around midnight on October 24, 2015, just months after graduating from SU, her son was in a horrific automobile accident that left Lehr with traumatic brain injury, a shattered shoulder and ribs, and a damaged spleen—among multiple other injuries.

Even today, he has no memory of an entire month spent in various hospitals; his first clear recollection was of Southwestern President Burger visiting him and his family for Thanksgiving dinner and playfully quizzing him on basic math questions. All told, Lehr’s full recovery took six months of surgery, rehabilitation, and physical therapy. “This happened to me, but it feels like it happened to someone else,” he says. Luckily, Lehr had “amazing family support,” and because he had equally supportive colleagues, he was able to return to work at the University of Texas Center for Transportation Research: Network Modeling Center (NMC), where Lehr had been working as a researcher focusing on traffic patterns since graduating from Southwestern. Even while battling the persistent side effects of the accident, Robert learned to code in R and Python and engaged in data visualization and modeling at NMC—despite being the only person on staff who didn’t already have a master’s degree or higher certification.

So being able to travel to Denver to accept his awards at MathFest 2018 was something of a personal triumph for Lehr. And his mother eventually agreed with him that the conference experience is a great segue into graduate school. This fall, Lehr will begin pursuing his master of science in sustainable design at the School of Architecture at the University of Texas at Austin. He attributes his interest in architecture in part to an interdisciplinary course he took at Southwestern with Associate Professor of Physics Mark Bottorff, Math Methods in Physical Science. “I admired [the course] for how it was taught,” Lehr says. He remembers Bottorff assigning an open-note, open-book assignment of six questions, which taught Lehr the value of taking good notes.

Futumura’s geometry class—the one from which the published article sprung—was another favorite SU course that influenced his decision to study architecture. In grad school, he’s looking forward to learning more about the theories of form and function, and the sustainable design degree will have applications beyond designing buildings. “It’s the intersection of mathematics and art, both passions of mine,” Lehr says. “It’s about having fun, creative, artistic ideas as well as wishing to make a practical difference in life… . I want to appeal to people’s moods, emotions, desires. I want to make them feel comfortable as well as provide practical utility.”

And just like his collaboration with Futumura on the viewpoint article, Lehr sees the UT program as a place where he can develop stronger architectural foundations for his myriad ideas about optimizing art and design: “I like to reimagine things—just standards that I look at and think, ‘That works, but can it be better?’ I have my own perspective, but I would like to educate myself on what’s better and having backing for that.” If the solution to the viewpoint question Lehr proposed in his article with Futumura is any indication, he will have no trouble creating designs that marry form with function in an elegant, innovative way.