Using Epsilon Hulls to Characterize and Classify Totally Disconnected Sierpiński Relatives

Taylor, T. L.

doi:10.1142/s0218348x15500152

Fractals

2015

DOI: 10.1142/s0218348x15500152

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Using Epsilon Hulls to Characterize and Classify Totally Disconnected Sierpiński Relatives

T. L. Taylor

Abstract: This paper presents an investigation of the use of -hulls to characterize and classify the totally disconnected Sierpiński relatives. These fractals all have the same fractal dimension, and the same topologies, but the -hulls provide ways to distinguish between such relatives. Examples of specific relatives are presented to illustrate the theory.

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Cited by 3 publications

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Equations and Interval Computations for Some Fractals

2018

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Very few characteristic functions, or equations, are reported so far for fractals. Such functions, called Rvachev functions in function-based modeling, are zero on the boundary, negative for inside points and positive for outside points. This paper proposes Rvachev functions for some classical fractals. These functions are convergent series, which are bounded with interval arithmetic and interval analysis in finite time. This permits to extend the Recursive Space Subdivision (RSS) method, which is classical in Computer Graphics (CG) and Interval Analysis, to fractal geometric sets. The newly proposed fractal functions can also be composed with classical Rvachev functions today routinely used in Constructive Solid Geometry (CSG) trees of CG or function-based modeling.

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Equations and Interval Computations for Some Fractals

2018

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Convex Hulls of Sierpiński Relatives

Taylor

Rowley

2018

Fractals

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This paper presents an investigation of the convex hulls of the Sierpiński relatives. These fractals all have the same fractal dimension but different topologies. We prove that the relatives have convex hulls with polygonal boundaries with at most 12 vertices. We provide a method for finding the convex hull of a relative using its scaling and symmetry properties and present examples. We also investigate the connectivity properties of certain classes of relatives with the same convex hulls.

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Bar-Codes of Sierpiński Relatives With Triangle Convex Hulls

Taylor

Rouse

2022

Fractals

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This paper presents results about the triangle Sierpiński relatives. These are Sierpiński relatives that have the same convex hull (whose boundary is a right isosceles triangle) as the Sierpiński gasket. In general, the Sierpiński relatives all have the same fractal dimension but different topologies. The special subset of triangle relatives includes are all both path-connected and multiply-connected. We investigate the epsilon hulls of the relatives (sets of all points within a distance of [Formula: see text] to the relative) to characterize and compare. This analysis includes the topological bar-codes which convey information about the connectivity of the [Formula: see text]-hulls of the relatives as [Formula: see text] ranges over the non-negative reals. We also prove that the growth rate of holes in the [Formula: see text]-hulls is equal to the fractal dimension.

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Using Epsilon Hulls to Characterize and Classify Totally Disconnected Sierpiński Relatives

Cited by 3 publications

References 8 publications

Equations and Interval Computations for Some Fractals

Equations and Interval Computations for Some Fractals

Convex Hulls of Sierpiński Relatives

Bar-Codes of Sierpiński Relatives With Triangle Convex Hulls

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