Harry Seldon's blog

Fractals, Chaos, and Control Systems on Rails


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.

Posted in , | 9 comments | Tags , , , , , , | atom



Use the following link to trackback from your own site:

  1. Social comments and analytics for this post From uberVU - social comments
    This post was mentioned on Twitter by fractalharry: Controls vs Chaos, a simple illustration: A visualization of chaos is given by fractals. I showed you the pictures... http://bit.ly/ckp6YV
  2. 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 : [http://events.polytechnique.fr/accueil/hommage-a-benoit-mandelbrot/] Ti...
  3. 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 : http://events.polytechnique.fr/accueil/hommage-a-benoit-mandelbrot/ Titl...


Leave a response

  1. KF on February 14, 2010 at 10:00 pm

    Great description. I think we are on the same page, but people often have different ways of explaining to themselves/visualising concepts like these.

    I understand it as the mix between order and randomness, structure and chaos. In everything, you have stochastic natural processes being guided by certain control mechanisms which limit the degrees of freedom a system has available to it. Control mechanisms can take many forms - the constraints of our physical bodies, our intentions, etc.

    In the same way we can’t easily stop a river from flowing (for water will always find a way), we can’t stop natural processes such as life from happening (this is sometimes referred to as the arrow of time). All we can do is direct the flow in ways which hopefully turn out to be beneficial.

    Thanks for the great post :)

  2. Harry Seldon on February 14, 2010 at 10:53 pm

    @KF “All we can do is direct the flow in ways which hopefully turn out to be beneficial.” Agreed. looks indeed like we are on the same page. Thx for your comment.

  3. Ken Gillgren on February 15, 2010 at 11:48 pm

    Thanks, I’ve got to say your presentation is spectacularly refreshing and clear. And now I understand why we tend to “overthink” complexity and end up creating high-maintenance, complex control algorithms.

  4. tcrouzet on February 17, 2010 at 02:19 pm

    Clear and clever… Bravo!

  5. Harry Seldon on February 17, 2010 at 11:25 pm

    Thx Ken and Thierry. Glad if I could come up with a clear post as it is complex to make things look simple ;-)

  6. Pat on July 21, 2011 at 11:21 am


    I’m doing some work at university on understanding the relationship between chaos and order in urban environments. The pictures of pruned trees is a fantastic way to illustrate this, thanks for some inspiration.

  7. Harry Seldon on July 25, 2011 at 10:30 pm

    @Pat, I am glad this article could inspire you. Gardens are, almost by definition, a good definition of shaping the nature, meaning ordering the chaos. With this respect, japanese gardens are amazing. They need a lot of work but at the end, in some sense, the garden should be seen as untouched nature. So, somehow, it can be opposed to the French gardens where the manprint is plain obvious.
    But finally, both types of garden are a way to control the nature.

    Best, H

  8. Matthew T Rader on September 29, 2012 at 09:06 pm

    I’m curious as to why you call nature’s algorithms simple compared to artificial ones? Would nature’ algorithm be DNA? If so, isn’t DNA far more complex than any algorithm written by a human or a machine?

  9. Harry Seldon on October 06, 2012 at 02:31 pm

    Nature’s algorithmes are simple because they are fractal. A fractal algorithm is simple because it is an iteration over a simple action. To take the example of the tree it is: take a branch that separates into 2 branches and iterate. You can replace 2 by a random number at each step to add some randomness. H

Leave a comment

Markdown enabled



Recent Comments

Recent Posts


actuators aircraft atc blog chaos chaos_theory charts control controllers controls crisis economy finance flight fractals git gnc gs guidance linux mandelbrot marketing navigation ns ofc on pilot rails ruby sas scs sensors statistics systems techcrunch thinkosphere tutorial typo ubuntu wifi