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Bar-Codes Of Sierpiåƒski Relatives With Triangle Convex Hulls

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  • T. D. TAYLOR

    (Department of Mathematics and Statistics, St. Francis Xavier University, Antigonish, Nova Scotia, Canada B2G 2W5, Canada)

  • S. ROUSE

    (Department of Mathematics and Statistics, St. Francis Xavier University, Antigonish, Nova Scotia, Canada B2G 2W5, Canada)

Abstract

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 𠜀 to the relative) to characterize and compare. This analysis includes the topological bar-codes which convey information about the connectivity of the 𠜀-hulls of the relatives as 𠜀 ranges over the non-negative reals. We also prove that the growth rate of holes in the 𠜀-hulls is equal to the fractal dimension.

Suggested Citation

  • T. D. Taylor & S. Rouse, 2022. "Bar-Codes Of Sierpiåƒski Relatives With Triangle Convex Hulls," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(09), pages 1-17, December.
    • Handle: RePEc:wsi:fracta:v:30:y:2022:i:09:n:s0218348x22501699
    DOI: 10.1142/S0218348X22501699
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