Thousands of Fractal Images

Why is geometry often described as 'cold' and 'dry'? One reason lies in its inability to describe the shape of a cloud, a mountain, coastline, or a tree. Clouds are not spheres; mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

Benoit Mandelbrot
The Fractal Geometry of Nature

Fractals are important, not because they have solved vast open problems in mathematics, or helped build a better mousetrap, or unlocked the secrets of rotating black holes. They are important because they suggest that, out there in the jungle of the unknown, is a whole new area of mathematics directly relevant to the study of Nature.

Ian Stewart
The Problems of Mathematics

The word fractal was derived from the latin fractus, meaning broken, by Mandelbrot (1975, Les objects fractals: forme, hasard et dimension), who gave a 'tentative definition' of a fractal set as a set with its Hausdorff dimension strictly greater than its topological dimension, but he pointed out that the definition is unsatisfactory as it excludes certain highly irregular sets which clearly ought to be thought of in the spirit of fractals. . . . Many of the classical fractal sets are 'self-similar', built up of pieces geometrically similar to the entire set but on a smaller scale.

K.J. Falconer
The geometry of fractal sets