Problem Sheet 10 MT1100 From Euclid to Mandelbrot Not marked: Solutions on Moodle later this week 1. Why is the Sierpinski gasket (the first few stages of its construction are below) a self-similar fractal? Calculate its fractal dimension. 2. The Menger sponge is constructed from a solid cube by repeatedly removing smaller cubes: the picture below shows the first four stages of the process. Cal- culate the fractal dimension of the Menger sponge. 3. The Koch snowflake is constructed from an equilateral triangle with side length 1 as in the diagram below, by adding infinitely many equilateral triangles of smaller and smaller size. (The perimeter of a Koch snowflake is made from Koch curves, as discussed in the lectures.) Show that the area of the equilateral triangle in Stage 0 is v3/4. Hence calculate the area a of the Koch snowflake. What is the length l of the perimeter of the Koch snowflake? 1
4. The middle fifth Cantor set is defined in the same way as as the standard Cantor set, except that the middle fifth of each interval is removed at each stage. (So in the first stage the interval (2/5, 3/5) is removed; in the second stage the intervals (4/25, 6/25) and (19/25, 21/25) are removed.) Find the total length Lj of the intervals that remain after j stages. Show that the fractal dimension of the middle fifth Cantor set is approximately 0.756471. Is this value consistent with the lengths Lj you have calculated? 2