An example of a curve having fractal dimension. Starting with an equilateral triangle, the middle third, PQ, say, of a side is replaced by the two lines PR and RP so that P, Q, and R form the vertices of a smaller equilateral triangle, with R outside the original enclosed region. This process is repeatedly applied to each line segment. The resulting ‘curve’ is the snowflake curve and has infinite length but encloses a finite area. Its fractal dimension is defined to be ln 4/ln 3, since each ‘edge’ of the curve contains four copies each of 1/3 size.
Snowflake curve. This is an example of a curve with fractal dimension ≠ 1.
Subjects: Probability and Statistics.