Chapter 24 - Sierpinski fractals and GCDs

Abstract

The Sierpinski triangle is a well known fractal that appears as a triangle with a triangular central region missing and smaller triangles removed from the remaining ones. The Sierpinski triangle has been described years ago as a limit of curves. It has also been constructed and generalized using simple bit operations and modular arithmetic. It has been often viewed as the attractor of an iterated function system. This chapter connects the construction of the Sierpinski triangle with other classical fractals. One of these is the “Sierpinski carpet” that has missing squares instead of triangles. The Sierpinski carpet also has interesting topological properties. Likewise, three- and higher dimensional fractals related to the Sierpinski triangle and carpet are also discussed, in a way consistent with the two-dimensional case. Among these three-dimensional generalizations are the Menger sponge and the Sierpinski tetrahedron. The fact that these fractals are constructed in a similar fashion is made evident by focusing that discrete versions of all these arise using inner products involving greatest common divisors and least common multiples on matrices involving base two and three addresses. These constructions admit generalization to arbitrary dimension and base.