Geometric Spirals from Convex Combinations (2012)

Undergraduates: Kiri Sunde, Jessalyn Bolkema

Faculty Advisor: Mark McCombs
Department: Mathematics

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Given n ≥ 3 points P0,P1,…,Pn−1 in R², we examine piece-wise linear spirals generated by taking convex combinations of m of those points where m ∈ {2,…,n}. In particular, for each k ≥ 0, let Pn+k = t1Pk+t2Pk+1+···+tmPk+m−1 where 0 ≤ t1,t2,…,tm ≤ 1 and t1+t2+···+tm = 1. This generates a sequence of points {Pk}∞k=0. For each k ∈ N, let Qk = Pk − Pk−1 be the vector connecting consecutive points. For each choice of n, m, and t1,…,tm, we establish necessary and sufficient conditions on the starting points P0,P1,…,Pn−1 that result in the lengths ||Qk|| forming a geometric sequence. We call the resulting piece-wise linear spirals geometric spirals. We prove several properties of geometric spirals including the fact that the geometric ratio is the modulus of an eigenvalue of a certain naturally arising matrix and that the angle between consecutive Qk's is constant and calculable from the starting points. This work generalizes work done by two earlier REU groups on the m = 2 case. Our research culminates in the following theorem: Given a spiral in<="" p="" style="box-sizing: border-box;">