BAR-CODES OF SIERPINSKI RELATIVES WITH TRIANGLE CONVEX HULLS

This paper presents results about the triangle Sierpinski relatives. These are Sierpinski relatives that have the same convex hull (whose boundary is a right isosceles triangle) as the Sierpinski gasket. In general, the Sierpinski 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 epsilon to the relative) to characterize and compare. This analysis includes the topological bar-codes which convey information about the connectivity of the epsilon-hulls of the relatives as epsilon ranges over the non-negative reals. We also prove that the growth rate of holes in the epsilon-hulls is equal to the fractal dimension.