Ashley Reiter remembers being involved in math competitions from a young age, such as the MATHCOUNTS competition in sixth grade, but she also remembers moments of being inspired simply by the beauty of math itself.
“I remember the first time I saw the proof that there were different sizes of infinity—I think I was probably in junior high then—and I just loved it. I thought, this is so beautiful.”
The daughter of mathematicians, Reiter (now Ahlin) left home for her last two years of high school to attend the School of Math and Science in Durham, NC, a few hours away. There, she took advanced math courses and began working on a research project that took first place at the 1991 Westinghouse (now Intel) Science Talent Search competition.
Pascal's Triangle
“I was really homesick at the beginning and would not have chosen to go away for school. But it was the first time that I’d really had peers, in the closest sense of the word, and been challenged together with other high school students,” says Reiter. “My senior year, I took Multivariable Calculus, and 5 of the 16 girls on my hall were taking Multivariable Calculus with me.”
For her winning project, "Fractals Generated by Pascal’s Triangle and Its Multidimensional Analogs," Reiter used number theory and analysis to calculate the dimensions of various fractals in not-so-obvious space.
Fractals — or geometric shapes with parts that resemble a smaller version of the whole — appear in nature and mathematics as the result of an iterated pattern. Everyday objects that are fractal-like include snowflakes, broccoli, and cloud formations. In math, patterns such as Pascal’s triangle (the rows of which begin with the famous sequence: 1, 1-1, 1-2-1, 1-3-3-1,…) can produce fractals from particular patterns within them. For example, Sierpinski’s triangle results from coloring in only the odd or only the even numbers in Pascal’s triangle.
Sierpinski's Triangle
Reiter took Pascal patterns further. “I then looked at what happens if you look at just the entries that are divisible by any particular number, prime or not prime, or power of a prime, or product of primes, or whatever. Then, also, I looked at the multidimensional versions of those, the coefficients of expanding something bigger, showed that all of those were fractals, and then figured out the dimension of those fractals.”
Unlike dimensions we are used to seeing — lines in one dimension, planes in two, space in three — fractals exist in fractional dimensions: e.g., inches to the 1.5 power rather than inches or inches squared. The aim of Reiter’s project was to figure out the units of dimension for these complex fractals.
After winning the Westinghouse STS, she attended Rice University and then went on to get her PhD in mathematics from the University of Chicago. She taught at the Maine School of Science and Math as well as at Vanderbilt University, and is now in Michigan with her husband and three children. She still teaches math online, via the Art of Problem Solving, and is working in her local community to develop elementary math opportunities for enrichment of young kids with talent. Additionally, she has volunteered her time teaching English as a Second Language and plans to work with international students more in the future.
Says Reiter, “People often ask if I want to go back and teach at the university level. I think probably not, for a number of reasons, one being that I just am really enjoying working for Art of Problem Solving, and enjoy working with bright young kids a lot.” Not to mention the work of taking care of three children while doing all this. “To take good care of a family is a high calling, too.”