Math 459, Senior Seminar
11/7/11
Name: Melinda Wiles
Title: Charm Bracelets
Source: Jon Froemke and Jerrold W. Grossman, An Algebraic Approach to Some NumberTheoretic Problems Arising from Paper-Folding Regular Polygons, Amer. Math Monthly, Vol.
95, No. 4 (Apr., 1988), pp. 289-307
Senior project ideas:
1) The paper is actually revisiting a topic written by Peter Hilton and Jean Pedersen. The graph
of f2 that Froemke and Grossman use as a visual representation replaces the “symbol” that Hilton
and Pedersen use. You could research the differences that these cause while proving the same
theorems. If this proves too easy, choose one visual representation of f2 and try to find as many
other situations to use that visual representation. I figure that using Hilton and Pedersen’s symbol
would prove to be a bit more interesting.
2) The authors mention that theorem 2 can be proven with a generalization of Gauss’s Lemma
due to Emma Lehmer. Compare and contrast the differences between the proof presented in the
paper and Gauss’s Lemma. This shouldn’t be too hard. If this is just too simple try to find
alternate ways to prove other theorems within the paper.
3) In their paper, the authors mention that the motivation behind the formation of the functions ft
and gt is similar to the Collatz problem, but they have found no nothing in this work that is
relevant to the problem. Try researching for possible connections between this problem and the
work within the paper. I have no idea how hard this would be, but considering the paper was
published in 1988, other papers may have considered this since then. If you do fail to find any
connections consider why nobody has made any.
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