An IFS for the Stretched Level-n Sierpinski Gasket

Abstract

Broadly, fractals are sets that exhibit a repeating pattern at multiple scales. One important fractal set is the Sierpinski Gasket (SG) which is made up of nested equilateral triangles. A variation of the classic Sierpinski gasket is to create n-levels with the equilateral triangles. Another variation is to stretch the points of intersection for the triangles in SG into line segments of length 0 <  α  < 1/3. When one combines these variations, one arrives at the stretched level-n Sierpinski gaskets (SSGⁿ) which are the focus of this work. We give an introduction to iterated function systems (IFS) and determine an IFS which generates SSG³. We then describe how one can acquire the IFS for SSGⁿ in general, and conclude with a theorem which determines the Hausdorff  dimension for SSGⁿ.