Page "Fractal" Paragraph 7

This also leads to understanding a third feature, that fractals as mathematical equations are " nowhere differentiable ".

In a concrete sense, this means fractals cannot be measured in traditional ways.

To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure.

But in measuring a wavy fractal curve such as the one in Figure 2, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.

This is perhaps counter-intuitive, but it is how fractals behave.