Math Circles on Fractals

UW shield

Winter 2005

University of Waterloo - Waterloo, Ontario, Canada

Homework 1

  1. Find the area of nth stage of the construction of the Sierpinski carpet, starting with a unit square. The following figures show the first four stages. What is the ultimate area of the Sierpinski carpet?

  2. Find the length of the square Koch curve and the area it encloses at the nth stage. The curve starts with four unit lengths that form a square. At the nth stage, each side of length 1/4n-1 is replaced by eight sides of length 1/4n as follows.

    (The result is one of the fractals on Paul Bourke's fractal web page.)

  3. [For experimentation] What could happen to the triangular Koch curve if each replacement segment could be pushed out on an arbitrary side? For example:

Back to Math Circles on Fractals        This page was last updated on 9 February 2005 by W. Gilbert