Understanding Fractality: A Polyhedral Approach to the Koch Curve and Its Complex Dimensions

Abstract

We extend our results about the Weierstrass Curve to the Koch Curve and provide exact expressions of the volume of polyhedral neighborhoods for the sequence of prefractal graphs which converge to the Koch Curve. We also introduce the associated local and global polyhedral fractal zeta functions. The actual poles of the global polyhedral fractal zeta function, which are all simple, yield the set of exact Complex Dimensions of the Koch Curve, a result which had never been obtained before.

Keywords

Koch curve prefractal approximations iterated fractal drum complex dimensions of an iterated fractal drum box–counting (or Minkowski) dimension fractal tube formula effective local and global tube zeta function effective local and global distance zeta function

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