MA4527 Computational Geometry Assignment 0 1. Consider the set C obtained from the interval [0, 1] by first removing the middle third of the interval and then removing the middle fifths of the two remaining intervals. Now iterate this process, first removing middle thirds, then removing middle fifths. The set C is what remains when this process is repeated infinitely. What is its fractal dimension? 2. Consider the fractal generated by replacing a line segment with the smaller segments shown in the following figure, where each new segment is exactly one-third as long as the original. Draw carefully the next two iterations of this process. What is the fractal dimension of the resulting fractal? Find the iterative function system for the generation of the resulting fractal. Figure 3. Compute exactly the area of the Koch snowflake. 4. The Sierpinski right triangle is generated by the following contractions: . 1 0 + 1 -y x 2 1 = y x A 0 1 + y 1 – x 2 1 = y x A y x 2 1 = y x A 2 1 0 (i) To which point does the sequence y x A A 0 0 n 1 2 2 converge? (ii) Show that the sequence y x )A A( 0 0n 01 accumulates on the two points . 0 3/2 and 0 1/3 (iii) Show that the sequence y x )A A A( 0 0n 012 accumulates on the points . 7/1 7/4 and , 2/7 1/7 , 4/7 2/7
