There are many applications of geometry in contemporary life — from the cartography used in GPS devices to the special effects used in movies.
“The little trick of why Gandalf looks much larger than Frodo in ‘Lord of the Rings’ in the same scene is a good example of the affine perspective, as our eyes perceive the smaller size,” said Bogdan Suceavă, professor of mathematics.
Matthew Rathbun, assistant professor of mathematics, further pointed out how affine geometry can be used: “Affine transformations preserve some geometric properties, while changing others. Imagine you have to run a mile, but it doesn’t matter where you do it. Wouldn’t you rather run on flat terrain in a mild climate than up a steep grade or through a desert, all other things being equal?
“Our research is about how you can transport challenging geometric demands to the nicest terrain possible, making your task much easier. But more important than solving the particular problems nicely, it’s about the bigger idea of looking for ways to make any problem easier by looking for what it has in common with other easier problems.”
Suceavă, Rathbun, Adam Glesser, associate professor of mathematics, and mathematics alumna Isabel Serrano co-authored the article, “Eclectic Illuminism: Applications of Affine Geometry,” published in The College Mathematics Journal, a publication of the Mathematical Association of America. The journal is designed to enhance classroom learning and stimulate thinking regarding undergraduate mathematics.
For their exceptional work, the Titan mathematicians received the 2020 George Pólya Award from the Mathematical Association of America, the world’s largest community of mathematicians and students. The national award, announced July 27, is a prestigious honor given for articles of expository excellence.
“Being chosen for the Pólya Award is a tremendous honor because it recognizes work for being mathematically interesting and of the highest expositional quality,” Glesser said. “This award also helps to vindicate the teacher-scholar model that so many CSUF faculty employ.”
The award, Suceavă added, conveys a culture of excellence in expository writing efforts over the decades by CSUF math faculty members.
“It’s something that we take great pride in. Our students come to Cal State Fullerton to learn and to find an environment where profound ideas are explained in an approachable way. This recognition at the national level is yet another proof that we are accomplishing this very idea,” he said.
This paper is truly a jewel. It presents very deep ideas, and it is likely to have a profound influence on anyone teaching Euclidean geometry. The writing is extremely clear and engaging, and the diagrams very helpful. This paper should be very readable by students and provides a nice selection of exercises to keep all readers entertained. — Pólya Award committee reviewers
The paper explains how techniques of affine geometry can be applied to geometry problems to make solving them easier.
“Math instructors familiar with affine techniques stand a better chance to solve intricate Euclidean geometry questions — that’s why such ideas are of interest to future K-12 teachers,” Suceavă explained.
The mathematicians’ goals with this research are to help high school and college students solve challenging geometry problems in math competitions, and to give aspiring high school teachers and college instructors ways to better teach linear algebra by using geometric examples.
“I hope that anyone who reads our article will be a little bit better at problem-solving in general. The idea of how to approach a problem, and how to relate it to special cases that are easier to understand is fundamental to critical and creative thinking,” Rathbun said.
Serrano ‘18 (B.A. mathematics-applied), a doctoral student in computational biology at UC Berkeley, worked on the research as an undergraduate.
“This work brings to light the importance of using different tools and perspectives to answer questions,” noted Serrano, who co-authored three other research articles with Suceavă, Glesser and Anael Verdugo, associate professor of mathematics.
“When we remain narrow-minded in our approach to problems, we may prevent ourselves from seeing ‘unconventional’ methods that simplify and make a problem approachable. Our paper wonderfully highlights the strength of keeping various mathematical techniques in mind and how this flexibility allows us to solve challenging problems.”