A Study of Sierpinski Fractals Based on Regular Polygons and Circles

Abstract

This paper studied the properties of Sierpinski fractals based on regular polygons and circles. Dilation is adopted as the basic principle to generate "perfect Sierpinski polygons" and "perfect Sierpinski circles"-defined as having neither overlap nor detachment among sub-shapes. The ideal contraction ratios for some regular polygons to become perfect Sierpinski polygons are calculated using trigonometry. A general formula is further derived by making a connection between perfect Sierpinski polygons and Sierpinski circles. A brief exploration is conducted of the Sierpinski circle model, from which some interesting properties are revealed, like the segregation of shapes. Based on these properties, a general formula to calculate the radius of the nth layer of sub-circles within one stage of iteration is derived. Lastly, the connection between the two Sierpinski models is discussed based on the limitations and implications of this study. Future researchers are recommended to dive deeper into the properties of Sierpinski travel and its relationship with other Sierpinski models.