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Fractals, Chaos, and Control Systems on Rails

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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:
controlled_trees

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:
chaos_vs_control

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
Garden with naturally shaped trees in the background
conic_trees
Trees pruned in a conic shape.
cubic_trees
Trees pruned in a cubic shape.

Feel free to leave your comments.

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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.

tree_snow_battle
Children making a snow battle under a magnificient fractal tree.

ghost_trees
Enchanting ghost trees.

forest_snow
Sceaux Castle Forest under the snow.

Sceaux_castle_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?
naked_tree_naked_young_lady
Wow, this naked young lady under the naked tree must be freezing. ;-)

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