Fractals and Multifractals, Introduction to

Fractals generalize Euclidean geometrical objects to non‐integer dimensions and allow us, for the first time, to delve into the study of complex systems, disorder, and chaos. In the words of B. Mandelbrot: “Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, nor does lightning travel in a straight line,” [1]. Indeed, much has changed in our perception of nature, and today it is hard to conceive of natural phenomena that are not fractal, in the same way that it is hard to conceive of everyday life dynamical systems that are not non‐linear. The discovery of fractals over three decades ago signaled a profound shift in the way we understand and analyze the physical world around us.

Not only does fractal geometry model complex disordered objects such as clouds and mountains, coastlines and lightning, but it also finds a beautiful new symmetry in the midst of all this complexity – invariance under dilation of space – and it is this self‐similarity...