What is Mandelbrot: Definition and 25 Discussions

Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM, where he became an IBM Fellow, and periodically took leaves of absence to teach at Harvard University. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences.
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 discovery of the Mandelbrot set in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order". His math and geometry-centered research career included contributions to such fields as statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy, and the social sciences.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. 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: Memoir of a Scientific Maverick, was published posthumously in 2012.

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  1. DaveC426913

    B The Mandelbrot Set has a third dimension: the Bifurcation Diagram

    This is wild. I was always fascinated with the Mandelbrot set, as well as the bifurcation diagram. I had no idea the Mandelbrot diagram was a different visualization of the bifurcation diagram. Question: is this video accurate? I always question the veracity of YouTube science videos.
  2. Arman777

    Python Mandelbrot Set: Code for High-Resolution Image to Zoom In

    Is there a code for the mandelbrot set that creates an image with an high resolution so that we can zoom and see the fractal over and over again ?
  3. CalcExplorer

    A How to Convert a Complex Logarithm to a Complex Exponential

    Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...
  4. E

    Are these permissible ways of writing Mandelbrot's Equation?

    When asked about his work, Mandelbrot wrote his equation as such: z -> z^2 + c Is it permissible to also write it as: z = z^2 + c and / or f(z) = z^2 + c
  5. W

    Investing in Stock Markets: Fat Tails, Persistence, Multifractals & Turbulence

    Does anyone here know much about these topics? I understand they surround the absence of normally distrubted returns, excessive kurtosis. Fat tails somehow disprove the EMH? Can anyone explain this argument? I've been advised that there are links to turbulence in fluid dynamics, joined by the...
  6. S

    The Mathematics of the Mandelbrot Set

    As a mathematician, what may you say are the beauties that you see in the Mandelbrot set??
  7. M

    Two exercises on complex sequences (one about Mandelbrot set)

    Homework Statement . I am trying to solve two exercises about complex sequences: 1) Let ##\alpha \in \mathbb C##, ##|\alpha|<1##. Which is the limit ##\lim_{n \to \infty} \alpha^n##?, do the same for the case ##|\alpha|>1##. 2) Let ##\mathcal M## be the set of the complex numbers ##c## such...
  8. MarkFL

    MHB Mandelbrot Monk: Ancient Fractal Geometry

    This is certainly not "breaking" news, but I saw this article a few years ago and happened upon it again, and wanted to share it here: THE MANDELBROT MONK It seems that what is perhaps the most iconic image of fractal geometry was known before its discovery by Benoit Mandelbrot. (Sun)
  9. P

    Find Equation of Main Cardioid in Mandelbrot Set & Minibrot

    Assuming that we could interpret the imaginary axis in the complex plane as the output of a relation, how would we find the equation of the curve that bounds the main cardioid of the M-set? Is there a way to find the equation of the main cardioid on a "minibrot" (e.g. if I zoom in on the fractal...
  10. K

    Visually attractive Mandelbrot zoom points

    Hi, I have been working on some Mandelbrot wallpapers for some time but have so far only visualized the whole set i.e. no zooming. I think that it could be nice with wallpapers visualizing a zoomed in part of the border of the set. Does anyone know coordinates to interesting places for zooming...
  11. C

    What is the best approach for plotting the Mandelbrot set using a computer?

    Homework Statement Note: This is not for homework. I'm trying to teach myself programming and this looks like a fun project. I want to plot the Mandelbrot set using a computer. Homework Equations Z_{n+1}=Z_n^2+c for some constant c The Attempt at a Solution Given: Z_0=0, maximum of...
  12. G

    Mandelbrot Set Matlab: Exploring & Zooming Around Points

    Hi! The Mandelbrot set is defined as a fractal set of points c of the plan for which the sequence defined recursively by: Zn+1 = Zn.^2c with z 0 = 0 does not tend towards infinity (in module). useful property If there exists an integer N such that | zN |> 2, then the sequence diverges...
  13. 1

    Mandelbrot Set Song - Catchy Math Themed Tune

    Anyone ever watch this? Catchiest math themed song I've ever heard. IE, it's actually goodish.
  14. Newtons Apple

    What is the significance of the Mandelbrot set and fractals?

    Can someone explain the Mandelbrot set to me? I know the equation is zn+1 = zn2 + c But what does this mean? Whats its basically saying here? All I know about it really is the existence of fractals, but why are fractals so mathematically important?
  15. A

    News RIP Benoit Mandelbrot: Father of Fractal Geometry Passes Away

    Not sure if others have heard, but Benoit Mandelbrot, the father of fractal geometry, passed away this week, as confirmed by the New York Times: http://www.nytimes.com/2010/10/17/us/17mandelbrot.html https://www.youtube.com/watch?v=ES-yKOYaXq0 And now the line "Mandelbrot's in heaven, at...
  16. C

    Understanding the mandelbrot set equation(s)

    I'm not sure if this is the correct section to be posting in. I'm writing a summary of the mandelbrot set and I'm not sure I understand how the points are calculated. I've got the equation: z = z^{2} + c This means each value is squared, and then a constant value c is added, to get a...
  17. V

    Show that -i is in the Mandelbrot Set

    1a: show w = -i is in the mandelbrot set show that -1-i is not in the mandelbrot set is w= -0.1226 + 07449i in the mandelbrot set, first show that z2 =0 don't know how to do any of them i tried, -(squareroot-1) ^2 = --1, + squareroot -1 idk
  18. T

    (any) Applications of Mandelbrot sets? Proof of fractal?

    Hi, What exactly is the importance of the Mandelbrot set in general? From what I've read, it seems more of a mathematical play thing than anything else.. there must be more to it than the disturbing pictures, no? Also, is there an easily understandable proof anywhere showing that the...
  19. S

    Quadratic Recurrence and the Mandelbrot Set

    I am currently studying the Mandelbrot set, and have a question about one of the statements on http://mathworld.wolfram.com/QuadraticMap.html" . It says the recurrence for the Mandelbrot set "is not in general solvable in closed form." What does this mean?
  20. rohanprabhu

    How is the Mandelbrot set plotted and how are the color values determined?

    I found this image on wikipedia: http://en.wikipedia.org/wiki/Image:Mandel_zoom_00_mandelbrot_set.jpg and i have seen this image like many times, but i have never understood as to how it is plotted. How is the value of the color decided? Plus, if it is a complex plane, how are the axes...
  21. S

    Mandelbrot Set- definitive point testing

    For those of you not familiar with the mandelbrot set, it is the set of all complex numbers c for which the following transform remains finite for an infinite number of iterations: z --> z^2 + c z is 0 for the first iteration My question is this: How can I conclusively determine...
  22. T

    Mandelbrot Set: Why Are Julia Sets Connected?

    Don't know if this is the right subforum, but here goes: The Mandelbrot set is formally defined as the values of a complex constant c for which the Julia Sets of the iterative function: f(z)=z^2+c are connected (i.e. consists of a single figure) Recall that the Julia sets are the launch...
  23. D

    Measure of the Mandelbrot Set

    What is the Lebesgue measure of the Mandelbrot set?
  24. H

    Exploring the Mandlebrot and Julia Sets: The Beauty of Fractal Patterns

    I am sure the same goes for you lot, I am fascinated by the complex patterns of fractals and recently found out it is generated by extremely simple algorithms (which takes weeks to run). What do I actually need run some algorithms that generates fractals?