Benoit Mandelbrot in the context of "Multiple citizenship"

A fractal landscape or fractal surface is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the surface resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.

Many natural phenomena exhibit some form of statistical self-similarity that can be modeled by fractal surfaces. Moreover, variations in surface texture provide important visual cues to the orientation and slopes of surfaces, and the use of almost self-similar fractal patterns can help create natural looking visual effects.The modeling of the Earth's rough surfaces via fractional Brownian motion was first proposed by Benoit Mandelbrot.

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length or perimeter. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus, the first systematic study of this phenomenon was by Lewis Fry Richardson, and it was expanded upon by Benoit Mandelbrot.

The measured length of the coastline depends on the method used to measure it and the degree of cartographic generalization. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be taken into consideration when measuring, and hence no single well-defined perimeter to the landmass. Various approximations exist when specific assumptions are made about minimum feature size.

The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/) is a two-dimensional set that is defined in the complex plane as the complex numbers ${\displaystyle c}$ for which the function ${\displaystyle f_{c}(z)=z^{2}+c}$ does not diverge to infinity when iterated starting at ${\displaystyle z=0}$, i.e., for which the sequence ${\displaystyle f_{c}(0)}$, ${\displaystyle f_{c}(f_{c}(0))}$, etc., remains bounded in absolute value.

This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.

In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern and tells how a fractal scales differently, in a fractal (non-integer) dimension.

The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale, see § Examples below.