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
MSC Classification: 11M41 28A12 28A75 28A80 Koch Curve prefractal approximations iterated fractal drum (IFD) Complex Dimensions of an IFD box-counting (or Minkowski) dimension fractal tube formula effective local and global tube zeta function effective local and global distance zeta function
MSC Classification: 11M41
28A12
28A75
28A80 Koch Curve
prefractal approximations
iterated fractal drum (IFD)
Complex Dimensions of an IFD
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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