Student Research Projects

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Undergraduate

The 2007 REU team, composed of Michael Cantrell and Robert Price, completed the project “Self-intersection properties of generalized Koch curves.” The students presented their results at the joint AMS/MAA meetings San Diego in 2008.

The 2006 REU team, composed of Rebecca Black, Phil Hudelson, Lisa Lackney and Jim Rohal, completed the project “Self-similar tilings of nilpotent Lie groups.” Jeff Adler was co-director of this project. The students presented their results at the joint AMS/MAA meetings in New Orleans in 2007.

Naomi Cummings Ducharme, a mathematics and secondary education major, completed an honors project on Penrose tilings.

Rebecca Brown, an applied mathematics major, completed an honors project entitled "Fractal Tilings in the Plane." Rebecca developed the necessary and sufficient conditions for a 2 x 2 matrix to be used to generate fractal tilings of the plane. The original 2 x 2 matrix M must be an integer matrix, but it may undergo a transformation (multiplication by a transformation matrix) in order to generate the desired tiling. Rebecca presented her results at the Research by Undergraduates poster session at the AMS/MAA meetings in San Antonio in January, 1999.

Sara Hagey, an applied mathematics major, completed an honors project entitled "Fractal tilings derived from complex bases." Sara studied the radix expansion of complex numbers in complex bases. She then used iterated function systems derived from the complex base to generate a tiling of the plane. Her paper explains the relationship between the radix expansions and the tilings generated. Sara presented her results at the Research by Undergraduates poster session at the AMS/MAA meetings in San Antonio in January, 1999.  This paper appeared in the Mathematical Gazette.

Jennifer Schenkenberger, a mathematics major with emphasis in education, completed a project to introduce fractal geometry into the secondary classroom. She has developed a collection of units on the chaos game, iterated function systems, the random iteration algorithm and other topics in fractals.

Matthew Palmer participated in a summer research program at the University of Akron, supported by the Stoller Funds and Buchtel College of Arts and Sciences.  He completed an honors project entitled “Irregular Sierpinski Triangles.”  Matt generated irregular Sierpinski triangles and analyzed the matrices used to construct the triangles. He associated with each irregular Sierpinski triangle a self-similar regularization.  The dimension of the regularization dictates the shape of the irregular Sierpinski triangle.  Matt presented his results at MathFest at UCLA in August, 2000.  His paper appeared in the journal Fractals in March 2004.

Miyuki Breen studied fractal tilings of the plane. She was able to determine the exact area of certain tilings.  Miyuki presented her results at MathFest 2001 at Madison, Wisconsin and also at the Nebraska Conference for Undergraduate Women. Her paper appeared in the Pi Mu Epsilon Journal in March 2003.

Maria Salcedo, a student at Youngstown State University, was selected to participate in the McNair Scholars program at UA. She used a replacement scheme to generate fractal tilings that demonstate some of the symmetries common to the wallpaper groups. Maria presented her results at MathFest 2003 in Boulder, CO and she received a Best Presentation Award. Her article will appear in Fractals.

Graduates

Mary Knust completed a thesis on “Integration on Fractals” where she developed a trapezoidal-type method for approximating integrals over fractal regions.

Samantha Fales completed a project on "Random Sierpinski Triangles" where she introduced elements of randomness into the construction of the Sierpinski Triangle.

Cathy Miller completed a project on "The Hausdorff metric and Julia sets." Cathy did a thorough study of the completeness of the Hausdorff metric and the application of that in the convergence of fractals. She also looked at the properties of Julia sets and the Mandelbrot set.