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:
Everything you have always wanted to know about fractals without daring to ask.
These two movies about fractals and chaos are amazing.
Thank you Prof. Mandelbrot for teaching us how to describe nature !
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.
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.
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!
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.
All parts follow.
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).
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. ;-)
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:
- Aims (requirements)
- A plan to reach them (control system)
- Means to act (actuators)
- Means to measure your effort (sensors)
- The ability to adapt to your environment and to what you have measured (back to the control system) Notice a good control system does allow you to change your aims as well as your plans along the road. Heading straight into a storm is probably not a good idea…
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.
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.
Let’s imagine that economy is an airplane. This airplane goes through a severe zone of turbulence, transforming the cruise in an heavily uncomfortable bumpy ride. Passengers start to complain, and feel more and more insecure. One of them finally calls the stewardess and asks: “What are the pilots doing ? Can’t they move us out of this unbearable situation ?” “Well… there is actually no pilot in this plane”, the stewardess answers politely. The passenger now gets really nervous, and says “What ??? Then we should run to the cockpit and try to do something before it is too late !”. The stewardess, feeling really sorry, replies with a “Uh, I’m afraid there is no cockpit in this plane, Mister”. And the story ends there, with the krach of the “economy” airplane.
I had read this little comparison a year ago already in a French newspaper. The author was Jacques Attali, a French economist. As a pilot, I have wanted to blog about this since that time.
To be a little more accurate, an analogy would assimilate companies to airplanes and market to Air Traffic Control (ATC). The problem is the market does not control anything. Imagine air traffic without ATC. Air space would be a mess. Aircrafts would collide all the time and basically nobody would trust air traffic. Finally nobody would take the plane. The air traffic system would be quickly dead. Yet, that is what is happening with the economy. Companies crash and collide because they have secant routes and nobody to help them. Economy is a mess. Most of the time ATC helps pilots by guiding them among the traffic and by giving them slots to take off and land. But sometimes ATC prevents pilots from going too fast to their destination because they would compromise safety of other airplanes. Unfortunately companies have no “market controllers” to talk to, hence the crisis.
Economy should learn 2 lessons from aerospace system engineering:
Systems must be controlled.
An airplane is highly unstable. It can fly because it is very actively piloted. Air traffic is unstable too, it is controlled by ATC. In the same way economy needs guidance, navigation and control. To say it differently economy needs regulation.
Systems must be robustified.
Even once an airplane is controlled and made stable there is still a lot of work to do to ensure that it is robust to all kinds of failures, even some non-anticipated failures.
Economy needs safety engineers. Currently there is only one safety engineer for the economy. He is Nassim Nicholas Taleb, the author of the Black Swan and Benoit Mandelbrot’s spiritual son. Else financial engineers behave more like terrorists than safety engineers. OK it is a bit harsh.
I began this post by the first point. So let’s develop now the second point.
An airliner has redundancy by design. It has 3 flight computers (and more). It has 3 fly by wire circuits. It has 3 hydraulics circuits. It has 2 wings (just kidding). It has 2 pilots. To design the airplane there is a whole team of engineers whose only role is to evaluate and improve the aircraft safety. They will verify that the airplane is as black swan proof (rare catastrophic event proof) as it can be. For instance, they will check that in the case of an engine explosion the projections won’t cut all the flight control wires. Obviously, propulsion engineer (in engine companies) will make sure an engine does not explode but, all the same, aerospace safety engineers do not take anything for granted. So they study the worst cases and they robustify the system. Moreover, they also robustify the system without any specific case of failure. For instance, the logic behind having several circuits is as simple as “something could happen”. You do not always need to know the exact failure scenario to robustify the system. Obviously, nothing (and no human pilot) being perfect, there are crashes. However, it is still safer to be in an airplane than in a car.
To come back to business, notice that safety by redundancy is the 6th point in the absolutely excellent article ”The Six Mistakes Executives Make in Risk Management” by Taleb. The mistake is “We are taught that efficiency and maximizing shareholder value don’t tolerate redundancy”. (A blogger sums up the six mistakes here).
If you are a manager, don’t leverage too much your department. Don’t think that each competency must be hold by only one person. Don’t make your best to reduce all redundancies. On the contrary identify your key activities and put redundancy on it. On the short term, reducing all redundancy gives you more profits but on the medium term it leaves you exposed to very easy failures (a sick person, an expert who leaves, etc.).
To conclude I will point out that Taleb loves to compare financial analysts to blind drivers. Here is one of his last sentence: “There were so many wise people around, but whom does Obama pick? The same people who were driving a schoolbus blindfolded, who have now been given a bigger bus.” You can find the citation in this article: Black swan now elephant in the room
Guidance refers to the questions ”where am I going to?”, ”How can the vehicle follow a trajectory?”. The trajectory itself is prepared by the Flight Planning System (FPS) (or Mission Planning System, MPS). During the flight, the Flight Management System (FMS) knows the trajectory and gives the current portion to the Guidance System (GS) . The Guidance System is in charge of converting the high level parameters (trajectory, waypoint positions) into a set of lower level orders that can be understood by the control, typically altitude, heading or directly a load factor, that is an acceleration. In some cases the guidance can have the role to compute a trajectory between 2 points.
It is also in charge of maintaining the trajectory and the other high level parameters. That means the guidance is itself a control loop. Typically the guidance is the Position Control System (PCS).
This is where things become tricky. Piloting and Guidance are essentially similar. The difference to keep in mind is that the pilot is a low level control loop whereas the guidance is a high level control loop. A nice world would be a world where guidance and pilot loops are two independent loops. Unfortunately both are coupled, and designing one, you need to keep the other one in mind.
Next chapter will be about GNC and human pilot.