(a) Consider the unit interval. A variation of the Cantor set is constructed by removing two line segments each of length 1 5 . Thus at stage 1, remove the segments between {1 5 . . . 2 5 } and {3 5 . . . 4 5 } from the unit interval, leaving three line segments remaining. Continuing in this way, construct the fractal up to stage 3 either on graph paper or on a computer screen. Find the length of segment remaining at stage k. Determine the fractal dimension of the mathematical fractal constructed to infinity.

(b) A Lévy fractal is constructed by replacing a line segment with a try square. Thus at each stage, one line segment of length, 1, say, is replaced by two of length 1 √2

. Construct the fractal up to stage 7 either on graph paper or on a computer screen. When using graph paper it is best to draw a skeleton (dotted line) of the previous stage. What is the true fractal dimension of the object generated to infinity?

(c) A Koch snowflake is constructed by adjoining the Kock curve to the outer edges of a unit length equilateral triangle. Construct this fractal up to stage 4 either on graph paper or on a computer screen and show that the area bounded by the true fractal A∞ is equal to A∞ =

2√3 5

units2 .

(d) The inverted Koch snowflake is constructed in the same way as in Exercise 1

(c) but the Koch curve is adjoined to the inner edges of an equilateral triangle. Construct the fractal up to stage 4 on graph paper or stage 6 on the computer.