Mathematics
Eric Chang
Lecturer
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Potential for UROP Funding
Overview
option 1: Generating fractals using Markov chains and DNA sequences
The term fractal was coined by Mandelbrot during the 1970’s to describe infinitely self-similar mathematical constructs that nevertheless show up – on a finite scale – throughout nature: coastlines, DNA, lightning bolts, and Romanesco broccoli are a few examples. The chaos game algorithm (CGA) was developed by Barnsley to construct fractals based on random input. The CGA takes a sequence of random inputs yet always outputs the Sierpiński triangle (or other polygons with modification). Instead of random inputs, one can use Markov chains – a key application of which is the Google page rank algorithm. In bioinformatics, Jeffrey extended the CGA by using DNA sequences in 1990; one inputs a string of DNA and the CGA spits out a fractal!
Students interested in this project should have some exposure to Linear Algebra. Some experience with programming is helpful but not required. Time will be allocated to learning the matrix algebra and Mathematica program.
option 2: Classifying Goldberg polyhedra made from PhiZZ units
The Pentagon-Hexagon Zig-Zag (PhiZZ) unit, invented by Tom Hull, is a modular origami unit. That is to say, one folds 30 of these, checks some diagrams, then assembles them into a dodecahedron. But one can also fold 36 of them into a polyhedron with 12 pentagon and 2 hexagon faces. Or 90 of them into a soccer ball, aka truncated icosahedron, aka Buckminster fullerene. Furthermore, each polyhedron has a “proper 3-edge coloring.” We will classify the first few levels of “Goldberg polyhedra,” plot their nets, and find their proper 3-edge coloring via Hamiltonian circuits. Only apply for this if you enjoy repetitive tasks.