For weeks, the math majors had been punching in equations, poring over the flood of numbers that their computers spit out, dissecting the patterns over and over on a room full of white boards, turning the boards around to find room for one more idea. All to solve a question about Pascal’s triangle.
The triangle is a widely useful number pattern, stacking numbers in rows to build a pyramid in which each number in a row is created by the numbers above it. Developed centuries ago, the visualization shows up all over mathematical literature, from coin-flip probabilities to the Fibonacci sequence. de Castro and students from across the West Coast had gathered in an office at Seattle University to spend their summer investigating a section of the pyramid: the rows with no entries that can be divided by a prime power.
"It was really exhilarating,” de Castro said. “It was fun to see all of this brain power being thrown at this board. Because that’s what it felt like, we were throwing it at this board. And it was cool to see it all come together.”
And now, after weeks of research, he and his colleagues’ work has been published in The Pi Mu Epsilon Journal, the publication for the national math honors society. The accomplishment means much to de Castro, who’s building a resume to someday share his excitement about mathematics as a professor.
“It’s amazing to be able to share this with whoever wants to read it. Whether they’re a mathematical audience or not, they can read it,” he said. “It’s interesting and it’s awesome to be able to share that.”
Before dissecting Pascal’s triangle, de Castro had met its cousin, of sorts: Eulerian numbers. Thanks to funding from the Barbara Ondrasik ’57 and David Groce Fellowship, de Castro gained a taste of what research is like—prior to leaving for Seattle—while investigating the Eulerian numbers’ behavior with Professor Mark Kozek. This year, he and Kozek are preparing that research for publication.
Kozek pointed out that besides the fellowship and research opportunity in Seattle, de Castro won a second fellowship—the Whittier Fellowship for Underrepresented Students in the Sciences—and also researched mathematics for a summer at University of California, Los Angeles.
De Castro always yearned for research opportunities, and after his experiences with the Eulerian numbers and Pascal’s triangle, he’s confident about what he wants out of his burgeoning career.
“This is the kind of research I want to do. It was exciting and exhilarating,” he said. “I’m going to get to do this one day for a living. I can’t wait.”
De Castro, who’s minoring in physics, plans on earning a Ph.D. in mathematics. He also has plans to volunteer with the Peace Corps to share the joy and beauty of mathematics abroad.
“It’s very beautiful,” he says of math. It’s interesting, cool, and powerful, for the sake of being interesting, cool, and powerful. It also challenges your assumptions and opens your mind to thinking clearly through an idea, he added. “That has helped me in my thinking about other topics and other ideas, and really, I see a beauty in it that I don’t see anywhere else.”
It’s also a universal language. de Castro felt amazed in Seattle to meet students from other schools and realize that, though they all learned mathematics in their own ways, they’re all speaking the same language.
“I can go to Washington State. I can go to Maine. I can go to another country and they’re teaching math there, and I can still do the same math,” he said. “Maybe the symbols are different, but once you know what the symbol is, we’re all talking about the same thing. And that was a mind-opening experience. And it was just fun to geek out with other kids for a whole summer about stuff that we all love together.”
De Castro and his fellow students who were involved in the research opportunity have also had a separate mathematics project published as part of a larger paper in Math Horizons, a magazine published by the Mathematical Association of America.
