## Mandelbrot, the Fractalist

Summer 2013 (8), 21st July, 2013

Benoit Mandelbrot, 2007

**Benoit B. Mandelbrot**(20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician, noted for developing a "theory of roughness" in nature and the field of fractal geometry to help prove it, which included coining the word "fractal". He later discovered the Mandelbrot set of intricate, never-ending fractal shapes, named in his honor.

As a child, his family fled to France in 1936 to escape the growing Nazi persecution of Jews. After World War II ended in 1945, Mandelbrot studied mathematics, graduating from universities in Paris and the U.S., receiving a masters degree in aeronautics from Caltech. He spent most of his career in both the U.S. and France, having dual French and American citizenship. In 1958 he began working for IBM, where he stayed for 35 years and was an IBM Fellow.

Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovering the Mandelbrot set in 1979. In so doing, he was able to show how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess" or "chaotic", like clouds or shorelines, actually had a "degree of order". His research career included contributions to fields including geology, medicine, cosmology, engineering and the social sciences. Science writer Arthur C. Clarke credits the Mandelbrot set as being "one of the most astonishing discoveries in the entire history of mathematics".

Toward the end of his career, he was Sterling Professor of Mathematical Sciences at Yale University, where he was the oldest professor in Yale's history to receive tenure.[9] Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, Institute for Advanced Study and Centre National de la Recherche Scientifique. During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography,

*The Fractalist*, was published in 2012.

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As mathematical equations, fractals are usually nowhere differentiable. An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.

The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word

There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time.Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature,technology and art.

**fractal**is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far". Fractals may be exactly the same at every scale, or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity*per se*to exclude trivial self-similarity and include the idea of a*detailed pattern*repeating itself.As mathematical equations, fractals are usually nowhere differentiable. An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.

The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word

*fractal*in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 21st century. The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin*frāctus*meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time.Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature,technology and art.

The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to formally define even for mathematicians, but key features can be understood with little mathematical background.

This idea of being detailed relates to

This also leads to understanding a

**The feature of "self-similarity",**for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus, the little man inside the head of the little man inside the head...). The difference for fractals is that the pattern reproduced must be detailed.This idea of being detailed relates to

**another feature**that can be understood without mathematical background: Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A regular line, for instance, is conventionally understood to be 1-dimensional; if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces.This also leads to understanding a

**third feature**, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring a wavy fractal curve, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. This is perhaps counter-intuitive, but it is how fractals behave.## Natural Phenomena with fractal features

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. Examples of phenomena known or anticipated to have fractal features are listed below:

- clouds,
- river networks,
- fault lines,
- mountain ranges,
- craters,
- lightning bolts,
- coastlines,
- Mountain Goat horns,
- animal coloration patterns,
- Romanesco broccoli,
- heart rates,
- heartbeat,
- earthquakes,
- snow flakes,
- crystals,
- blood vessels,
- pulmonary vessels,
- ocean waves,
- DNA,
- various
vegetables (cauliflower & broccoli),
- soil
pores,
- Psychological subjective perception.