| Dimension measures roughness, or visual complexity, or how well the object fills the
space it inhabits. Here we see a family of fractals, starting with a straight line (N = 4 pieces,
each scaled by r = 1/4), of dimension 1, and ending with a filled-in triangle (N = 4 pieces,
each scaled by r = 1/2), of dimension 2. In this family of fractals, we keep N = 4,
and step r from 0.25 to 0.5 in increments of 0.025. As their dimension increases, the Koch curve
relatives come closer to filling in a solid region of the plane |
