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Abstract

Consider the following inductively defined set. Given a collection U of unit magnitude complex numbers, and a set initially containing just 0 and 1, through each point in the set, draw lines whose angles with the real axis are in U. Add every intersection of such lines to the set. Upon taking the closure, we obtain R(U). We investigate for which U the set R(U) is a ring. Our main result holds when 1 is in U and the cardinality of U is at least 4. If P is the set of real numbers in R(U) generated in the second step of the construction, then R(U) equals the module over Z[P] generated by the set of points made in the first step of the construction. This lets us show that whenever the pairwise products of points made in the first step remain inside R(U), it is closed under multiplication, and is thus a ring.

Author Bio

Jackson Bahr is a fourth year mathematics major at Carnegie Mellon University. He expects to earn a Master's degree along with a B.S. in May 2017. He performed this research at Carnegie Mellon's SUAMI program in the summer of 2015. He enjoys playing trombone and bridge, and is currently interested in differential geometry.

Arielle Roth has a B.S. in Mathematics from Elizabethtown College. She completed this research during the 2015 SUAMI program at Carnegie Mellon University. Her mathematical interests include analysis and algorithms. However, in her free time she enjoys reading both science fiction and fantasy novels and exploring the logical models that exist in those fictional worlds.

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