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Fractals, Chaos, and Control Systems on Rails


Symposium : Tribute to Benoit Mandelbrot at the Ecole Polytechnique

Posted by Harry Seldon on February 28, 2011

Tribute to Mandelbrot

A symposium specific to a tribute to Benoit Mandelbrot will take place at the Ecole Polytechnique on the 17th and 18th of March 2011. Entrance is free but upon registration :


Title of the event is “Universalities and fractals”. You can find the absolutely fascinating program at the previous address.
Here are just a few presentations :

  • Heinz-Otto Peitgen. University of Bremen and Florida Atlantic University « The Mandelbrot Set: Revitalizing Iteration Theory and Popularizing Mathematics »
  • Luciano Pietronero, Universita La Sapienza, Rome « Fractal Cosmology »
  • Laurent Calvet, HEC Paris et National Bureau of Economic Research (USA) « Risque extrême et régularité fractale en finance » (Extreme risk and fractal regularity in finance)
  • Jens Feder, Physics of Geological Process, Université d’Oslo, “Fractals flow and Fracture”
  • Peter Jones, Yale University « Product Formulas for Measures and Applications to Analysis »

By the way of this blog, I much thank the organizing committee as it seems to me that Mandelbrot was one of the greatest genius of the XXth century. I have been much awaiting this kind of tribute in France. That’s why I will deeply regret for a long time the fact that there is no way I can come on the Friday.

Here is a reminder of the few articles I wrote about Mandelbrot:

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Best videos about fractals chaos and Benoit Mandelbrot

Posted by Harry Seldon on October 19, 2010

Everything you have always wanted to know about fractals without daring to ask.

These two movies about fractals and chaos are amazing.

In French and in only one part here

Thank you Prof. Mandelbrot for teaching us how to describe nature !

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Tribute to Benoit Mandelbrot

Posted by Harry Seldon on October 18, 2010

Benoit Mandelbrot is, imho, the biggest genius of the XXth century and our time. His work has already technologically changed our world (CGI, cell phone antennas, processors shapes, unerstanding stock market krachs, etc.). But the day, the philosophy, the epistemology, behind his work is understood, the day, the “language of nature” that Mandelbrot taught us, is understood, this day, the world will be a lot more peaceful.

Back in February 2009, I wrote that the celebrity I would love most to meet was Benoit Mandelbrot. I could partially fullfill that dream almost exactly one year ago (on the Sunday 24th of October 2009). While he was in Paris to present the amazing movie Fractals: Hunting the hidden dimension I got the privilege to talk to Benoit Mandelbrot on the phone. I have never posted about that because the talk was quite personal but I guess I will someday. For now, I can say for sure that Benoit Mandelbrot was a great man with an amazing kindness and a very nice sense of humor. He was so kind to tell me to call him after I sent him an email. And on the phone, when I asked him why he was not giving more conferences, he answered me : “you know, I am old, most people think I am already dead.”

Well before that encounter I grew up reading Mandelbrot’s books, among them “The Fractal Geometry of Nature”. Prof. Mandelbrot, I already miss you.

About memories, here is a best of the posts I wrote about fractals, Benoit Mandelbrot, chaos and controls. The first article was a short portrait of Mandelbrot where I campaigned for him to have the Nobel Prize in ALL categories : economy, physics, medecine (biology), chemistry, and even litterature and peace.

If you wish to pay a tribute to Benoit Mandelbrot, please, feel free to do so in the comments.

Children making a snow battle under a magnificient fractal tree.

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Benoit Mandelbrot, father of the fractal theory, has just died

Posted by Harry Seldon on October 17, 2010

Benoit Mandelbrot is leaving us way too soon, even at 86. There is still so much to do to educate people about fractals.

I got the incredible luck to speak to him on the phone almost exactly one year ago (on the Sunday 24th of October 2009). He was still in good shape and he had quite a lot of projects. Among them, he was preparing a book of his memories. I hope he had the time to finish it.

I am shocked by this news.
All my thoughts go to his familly.

In an interesting but sad coincidence that life can do, I met this current week Nassim Nicholas Taleb who is, in some sort, Mandelbrot’s disciple.

The king is dead. Long live the king.
Mandelbrot is dead. Long live Taleb.

For more information, see the links given by the message sent by the Finance and Mandelbrot Facebook group :

Philippe Herlin October 17 at 10:01am R.I.P. BENOIT MANDELBROT, 1924-2010

Finance & Mandelbrot

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Controls vs Chaos, a simple illustration

Posted by Harry Seldon on February 14, 2010

A visualization of chaos is given by fractals. I showed you the pictures of fractal trees taken during a walk at the Parc de Sceaux after a snowfall in Paris. But I had taken a few more pictures of the very beautiful French garden of this Park.
A French garden (“jardin à la française”) is a nice illustration of control. Basically, the gardener controls the shape of the trees. The proximity of the natural trees with their fractal shapes and the gardened trees allowed me to take great pictures that show this contrast between chaos and control.

Let’s begin with my preferred one:

I love the contrast between these cute spherically pruned trees and the majestic naturally shapped oaks behind. Notice that the apparent complexity of the fractal shape of the tree does not imply it is very difficult to prune a tree. It is more the size and the hardness of the branches that will decide how hard it is to prune the tree into a round shape.
Which tree shape is complex, the natural fractal shape or the artificial round shape? In terms of time, to accurately describe the shape, it is quicker to draw a circle than a fractal, so the circle is much simpler.
In terms of work to obtain the shape, it is the opposite. To get the fractal shape all you need to do is let nature do its job (with simple algorithms). To get a nice spherical shape you will need to prune the tree regularly. It is thus complex to get simple shapes.
Pruning a tree might not sound a complex process. However, the complexity comes from the fact the gardener will want to minimize his work on each tree. He will want to know the minimum frequency at which he needs to prune each of his trees. When in the season, which branch length (according to the tree halth), which tools, that makes many questions that complexify the control algorithm.
So yes gardeners as many other people do optimal control engineering without knowing it. Each time you ask yourself a question such as at which frequency should I do this, you are asking you the central question of control engineering. Too slow and you don’t get the performance you want, too fast and you overwork, you overconsume your energy.
A key factor of success is to do things at the good frequency. Unfortunately, this optimal bandwidth is complex to obtain.

To get back to the simple vs complex question, the key of the simplexity paradox lives in these points:

  • A simple natural algorithm leads to a complex shape.
  • A complex control algorithm leads to a simple shape.

More generally, we can sum this up like explained in this figure:

Notice that the line from nature to artificial world is continuous. I am not excluding mankind from nature.
Notice also that this separation between a chaotical world and a controlled world is very similar to Nassim Taleb’s separation between extremistan and mediocristan. I had already quickly talked about this description of extremistan and mediocristan as unstable (chaotical) and stable (controlled) systems in a previous post.

Then, here are a few more pictures of the snowed gardened.
Garden with naturally shaped trees in the background
Trees pruned in a conic shape.
Trees pruned in a cubic shape.

Feel free to leave your comments.

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How did we get here? Chaos vs God

Posted by Harry Seldon on January 18, 2010

I love the description given by the BBC for their documentary “The Secret Life of Chaos” (which you can watch here).
As I have written a few articles about fractals, chaos and controls lately, I have added links internal to this blog to the text.

“Chaos theory has a bad name, conjuring up images of unpredictable weather, economic crashes and science gone wrong. But there is a fascinating and hidden side to Chaos, one that scientists are only now beginning to understand. It turns out that chaos theory answers a question that mankind has asked for millennia - how did we get here?

In this documentary, Professor Jim Al-Khalili sets out to uncover one of the great mysteries of science -

  • how does a universe that starts off as dust end up with intelligent life?
  • How does order emerge from disorder?

It’s a mindbending, counterintuitive and for many people a deeply troubling idea. But Professor Al-Khalili reveals the science behind much of beauty and structure in the natural world and discovers that far from it being magic or an act of God, it is in fact an intrinsic part of the laws of physics.

Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern. The natural world is full of awe-inspiring examples of the way nature transforms simplicity into complexity. From trees to clouds to humans - after watching this film you’ll never be able to look at the world in the same way again.”

Notice that this introduction can be sum up by “Chaos vs God” or “Chaotical Design vs Intelligent Design”. However, anyway, one question remains: who created the laws of physics? Or how were created these laws of Physics, if you prefer ;-)

I have often privately said that James Gleick’s Chaos book gives clearer answers than the Bible about our world. Now is the time to say it publicly!

Enjoy the documentary!

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The Secret Life of Chaos (BBC 2010)

Posted by Harry Seldon on January 18, 2010

The BBC aired on Thursday, January 14th an excellent documentary about Chaos, Fractals and Nature. You can watch it right here thanks to YouTube. If you are in UK you can also watch it on the BBC website at this address.

I am glad the BBC helps making these subjects popular and fashionnable more than 20 years after James Gleick’s Chaos book.

Part 1

All parts follow.

Part 2

Part 3

Part 4

Part 5

Part 6


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The fractal Google logo

Posted by Harry Seldon on January 14, 2010

Thx Pixgeeks for reminding me of this nice Google logo involving fractals.


It was in memoriam to Gaston Julia’s Birthday.

In case I need to precise, the fractals you see on the logo are called Julia sets because the French mathematician Gaston Julia described them first. However, most of my readers already know that, right? ;-)

To say something only initiated people can understand: “The Mandelbrot set contains all Julia sets”. (That is why the fractal on the left is actually the Mandelbrot set.)

You can check all google logos here.

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Winter is the enchanting fractal season: Snow and Naked Trees

Posted by Harry Seldon on January 10, 2010

You probably already know that the snowflake and the tree branches are the canonical examples of fractals.

So, as in Paris we have the chance to have currently a lot of snow, I went to the “Parc de Sceaux” to make these wonderful pictures.
I only regret the sky was not as blue as in Normandy.

Children making a snow battle under a magnificient fractal tree.

Enchanting ghost trees.

Sceaux Castle Forest under the snow.

Sceaux Castle under the snow.
This picture will be the inspiration for a future post. Can you see why?

If you have pretty winter pictures you want to show, you are welcome to link to them in the comments.

Do not hesitate to contact me if for some reason you want the pictures in full size (5 MegaPixels).

Happy new fractal year!

PS: In the first picture the tree is not the only one to be naked, can you see the naked young lady in the picture?
Wow, this naked young lady under the naked tree must be freezing. ;-)

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Happy new fractal year!

Posted by Harry Seldon on January 10, 2010

Picture showing the fractal invariance of scale in a tree. Background is the Battle of Normandy (D-Day) Memorial, in memoriam to the allied forces who liberated Europe from the Nazi yoke, Caen, France.

I wish you to have all your wishes realized. But to be a little more accurate, I actually wish you to precisely know what you want and wish. Because wishes have a much better chance of becoming true if you can clearly formulate them.

That-is-to-say, in order to clearly know what you want, and how you can get it, you will need:

So, as you follow where I am heading to, yes I do wish you to be a full pilot of your own life, not a passenger. And do not forget “Goals are dreams with a deadline” as one says.


If you find your life is too much of a fractal it might be because you don’t control enough your life. It is normal to perceive the world outside of you as a fractal, with its good and bad news, with positive or negative black swan. But inside of you the way news affects you is very much under you control. The way you behave is under your control. World might be fractal, your mind might not if you master it.
All the fun of life is knowing what you can change and what you cannot.

To read more about the duality Fractals vs Control check this article about extremistan and mediocristan.

Be in control of the first thing you can readily control in that world: yourself ;-)

If you are interested in knowing more about how you actually control yourself, check the subject of NLP: Neuro-Linguistic Programmation.

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