Natura non facit saltum or, Nature does not make leaps… If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).

A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales.

Being a language, mathematics may be used not only to inform but also, among other things, to seduce.

For most of my life, one of the persons most baffled by my own work was myself.

I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid—a term used in this work to denote all of standard geometry—Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being “formless,” to investigate the morphology of the “amorphous.”

I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.

Many important spatial patterns of Nature are either irregular or fragmented to such an extreme degree that … classical geometry … is hardly of any help in describing their form. … I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals—or fractal sets.

Round about the accredited and orderly facts of every science there ever floats a sort of dustcloud of exceptional observations, of occurrences minute and irregular and seldom met with, which it always proves more easy to ignore than to attend to.

Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialties. The rare scholars who are nomads-by-choice are essential to the intellectual welfare of the settled disciplines.

The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.

The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient.

What is science? We have all this mess around us. Things are totally incomprehensible. And then eventually we find simple laws, simple formulas. In a way, a very simple formula, Newton’s Law, which is just also a few symbols, can by hard work explain the motion of the planets around the sun and many, many other things to the 50th decimal. It’s marvellous: a very simple formula explains all these very complicated things

Why is geometry often described as “cold” and “dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a 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… Nature exhibits not simply a higher degree but an altogether different level of complexity.

Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] … an historical role … in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor’s set theory and Peano’s space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded … as “pathological,” .… as a “gallery of monsters,” akin to the cubist paintings and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.
Now, as Mandelbrot points out, … Nature has played a joke on the mathematicians. The 19th-century mathematicians may not have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.

Most complex object in mathematics? The Mandelbrot Set, named after Benoit Mandelbrot, is represented by a unique pattern plotted from complex number coordinates. … A mathematical description of the shape’s outline would require an infinity of information and yet the pattern can be generated from a few lines of computer code. Used in the study of chaotic behavior, Mandelbrot’s work has found applications in fields such as fluid mechanics, economics and linguistics.