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Iterated Cyclic Averaging of Points

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Starting with a set of points in the complex plane (marked by red crosses: ×), iteratively form new points with where rescales and repositions the family . Even for random distributions of the initial , in the limit as , the points arrange themselves on smooth curves.

Contributed by: Michael Trott with permission of Springer (March 2011)
From: The Mathematica Guidebook for Numerics, second edition by Michael Trott (© Springer, 2008).
Open content licensed under CC BY-NC-SA

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For a closed‐form solution of the corresponding two‐term recursions, see

K. Mas'lanka, J. Cisło, Order from Randomness.

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