Page "Fractal" Paragraph 6

This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived.

A regular line, for instance, is conventionally understood to be 1-dimensional ; if such a curve is divided into pieces each 1 / 3 the length of the original, there are always 3 equal pieces.

In contrast, consider the curve in Figure 2.

It is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured.

The fractal curve divided into parts 1 / 3 the length of the original line becomes 4 pieces rearranged to repeat the original detail, and this unusual relationship is the basis of its fractal dimension.