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Published online by Cambridge University Press: 18 March 2010
Introduction
In mathematics, we are often faced with the problem of analyzing a sequence of real numbers xn. The sequence may come from a purely mathematical setting or from a computer output. What is the asymptotic behaviour of xn? Is it periodic, dense, uniformly distributed? What are the dynamics of the sequence? In this paper we discuss a method for ‘graphing’ xn so that such properties become more apparent.
Computer scientists use scattter plots to achieve this goal with some sucess. Our method, invoking fractal geometry, improves on scatter plots.
Given a sequence of real numbers xn there exists a Jordan curve Q in the plane called a fractal plot. The geometry of Q reflects analytical properties of xn.
We present in this paper the general algorithm for producing Q from xn. The sequence xn = nα (mod 1) where a is an irrational number is especally interesting and for such a sequence we call Q a continued fractal. We describe a dictionary for associating the geometry of Q with the analytical properties of xn. For example, we prove that if a is quadratic then Q is asymptotically self-similar.
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