Fractal Geometry. Fractal. A fractal is an object or quantity that displays self-similarity , in a somewhat technical sense, on all scales.
The object need not exhibit the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension . The prototypical example for a fractal is the length of a coastline measured with different length rulers . The shorter the ruler , the longer the length measured, a paradox known as the coastline paradox . Fractal. Figure 1a.
The Mandelbrot set illustrates self-similarity. As the image is enlarged, the same pattern re-appears so that it is virtually impossible to determine the scale being examined. Figure 1b.