MEGL Project Proposal: Visualizing the geometry of continued fractions Team: Anton Lukyanenko (faculty), 1 graduate student, 3 undergraduate students Apply by January 15, 2018 One way to write a number like pi is to specify a sequence of continued fractions that represent it. So pi would be approximated by 3, then by 3+1/7, then by 3+1/(7+1/15) and so on. One can study continued fractions involving complex numbers as well, which leads to pictures like the one below. Continued fractions have lots of great number-theoretic properties. Better yet, keeping track of the numerators and denominators leads to thinking about 2-by-2 matrices and then to (possibly complex) hyperbolic geometry. The goal of this project will be to illustrate the connection between hyper- bolic geometry and continued fractions — and then to see what new things we can prove about the two. Students working on the project don’t need to know about continued frac- tions or hyperbolic geometry, but should be ready to learn about both of those. Undergraduates: apply at http://meglab.wikidot.com/opportunities. Graduate students: contact Dr. Lukyanenko directly. 1
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MEGL Project Proposal: Visualizing the
geometry of continued fractions
Team: Anton Lukyanenko (faculty), 1 graduate student,3 undergraduate students
Apply by January 15, 2018
One way to write a number like pi is to specify a sequence of continuedfractions that represent it. So pi would be approximated by 3, then by 3+1/7,then by 3+1/(7+1/15) and so on. One can study continued fractions involvingcomplex numbers as well, which leads to pictures like the one below.
Continued fractions have lots of great number-theoretic properties. Betteryet, keeping track of the numerators and denominators leads to thinking about2-by-2 matrices and then to (possibly complex) hyperbolic geometry.
The goal of this project will be to illustrate the connection between hyper-bolic geometry and continued fractions — and then to see what new things wecan prove about the two.
Students working on the project don’t need to know about continued frac-tions or hyperbolic geometry, but should be ready to learn about both of those.
Undergraduates: apply at http://meglab.wikidot.com/opportunities.Graduate students: contact Dr. Lukyanenko directly.
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