Funny things can happen when we assume simple processes such as measuring the perimeter of a rectangle can be applied without thought to natural shapes, coastlines for example.
Standard approach The standard approach to measuring lengths of curves is to approximate the curve by straight line segments and add up the lengths of the segments.
Problems Problems with the standard approach: smaller measuring scales are sensitive to smaller features of the coastline.
Richardson Measuring the lengths of coastlines at different scales, Richardson observed behavior significantly different from that found when measuring smooth curves. For most coastlines, measured length appears to increase rapidly as the measuring scale is reduced.
Mandelbrot Mandelbrot's interpretation of the slope as a dimension.
Examples Some examples of coastlines
Sapoval Why are so many coastlines fractal? Bernard Sapoval suggested a reason, based on his study of fractal drums.