Fractals

How does one describe the skyline of a mountain range, the outline of a tree or a coastline, or the spatial structure of a snowflake? A description in terms of regular geometric shapes of Euclidean geometry, formed from straight lines and smooth curves, appears inadequate to capture the intricate structure of such objects. Benoit Mandelbrot introduced the word fractal to describe such "shapes made of parts similar to the whole in some way". For a start we may adopt the following definition. Definition of Fractal (informal): A fractal is a shape that appears self-similar on multiple spatial scales, that is, any piece of it looks like the whole after a change of scale (magnification).

The technical term that describes self-similarity of shapes under change of observation scale is scale-invariance. Systems that are scale-invariant do not have any characteristic length, that is a typical or mean length. For example, if one observes an aerial photograph of a coastline, it is difficult to guess the actual size of the features unless some man-made objects are also visible. This is because, as mentioned above, coastlines are self-similar on a wide range of scales, or approximately scale-invariant, while man-made products have a natural characteristic length: A car has a characteristic length of about $5$ m, a house about $10$m and Man has a characteristic length of $2$m. Objects that have a characteristic length scale look different at different magnifications: for example, the fingers, the arm, and the torso of a human body are not self-similar shapes.