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.

Part 2

Part 3

Part 4

Part 5

Part 6

Enjoy!

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. ;-)

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.

If you don’t like too much specialized books, you will like these ones because each one of them will speak about several topics among geology, economy, biology, social sciences, and climate.

Moreover, you will find some element of answers for several popular questions of our days like:

• How does the climate evolve?

• Why are we in the middle of a great economical crisis? (By the way, if you want good new about the crisis)

• Why politicians appear to be “all rotten anyway”?

• Is the key to success luck or hard work (or both)?

• Why species disappear?

• Why can’t we predict the weather nor the stock market?

• Can Chaos Theory be actually useful to something? (Chaos theory is also about knowing what you can’t predict exactly so that you can prepare for the worst)

So, let’s go to the point, here are these absolutely marvellous books. Notice, they are somehow sorted by order of importance.

• The Fractal Geometry of Nature, by Benoît Mandelbrot; W H Freeman & Co, 1982

• Chaos: Making a New Science, by [James Gleick] (http://www.around.com/)

• The Black Swan: The Impact of the Highly Improbable by Nassim Nicholas Taleb (If you have already read it, be sure you have read this post about mediocristan and extremistan)

• Simplexity: Why Simple Things Become Complex (and How Complex Things Can Be Made Simple)

• The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin and Reward.

• Fooled by randomness by Nassim Nicholas Taleb

• Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977

So, in case it was not clear, now you see why my Twitter name is @Fractalharry ;-)

You can propose other books in the comments.

Enjoy your reading.

PS I have linked to Amazon for your convenience but I don’t touch any commission!

So, is the tree shape complex or simple? Thanks to Mandelbrot we know now that the shape is a lot simpler than it seems. Associating the notions of predictability and simplicity, the converse is also true: it is more complicated than you could think even if you have heard of fractals. Hence this notion of simplexity, contraction of simplicity and complexity.

Here are a few good books on the subject:

• Simplexity: Why Simple Things Become Complex (and How Complex Things Can Be Made Simple)
• Chaos: Making a New Science, by James Gleick
• Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977
• In French, La simplexité by Alain Berthoz

If you know about fractals and chaos, you must be already familiar with that fact that simplicity can bring complexity quickly and easily. But you might not know this term of simplexity.
More generally, each time you think “this thing is a lot simpler than I had imagined at first”, you experience simplexity: in fact, you changed your first impression of overall complexity by discovering the underlying simple principles.

While we are at it. There is a field where simplexity shows all its magnificence: it is in finance. International finance looks complex but there are a limited number of principles behind it. You can even fairly easily model a stock chart. (Even if this model has nothing to do with the actual models used by financial analysts). But even with a good model you cannot predict easily the stock chart of a determined company.
For more info, you can have a look at the Facebook group “Finance & Mandelbrot”.

And at last, because I cannot prevent from saying it again, If you want to keep things simple, then regulate them. Contrary to what you could think, you do not need very accurate models to control a system.

It looks like main stream media are slowly understanding Mandelbrot’s work and they are speaking about it. I found yesterday an excellent article in one of the main French financial newspapers about Mandelbrot’s work on finance. The second good news, contained in that article, is that Mandelbrot’s book are being republished. A third good news, is that the author of the previous article, Philippe Herlin is very proactive and has just created a Facebook group called “Finance & Mandelbrot”. I joined immediately ! An instant interest of this group is that Philippe Herlin gives a list of news articles about the subject (given below). Please notice the link called How Fractals Can Explain What’s Wrong with Wall Street which is dated on February 1999. Yep, it is not a typo for 2009 !
Last but not least, there is a significant political news. Mandelbrot showed that the financial system was fractal, exhibiting the same behaviour at all scales. Actually that was the start of his long and brilliant career. Later on he showed the same behaviour for climate, geology, ecology, or biology. Actually about any natural system is fractal and even chaotical. However, there is a good way to smooth the behaviour of a system, it is called control. Our world needs more control (distributed control whenever possible). That is to say, more rules, more ways to enforce them and more ways to measure their efficiency. So the good news is today European leaders backed sweeping new regulations for financial markets. Questions remain, will the rules be good and will they be enforced?

By the way, if you want to understand a little better the whole world, if you want to hear about economy, climate, geology, ecology, or biology in less than 200 pages, just read these two amazing books. Your vision of the world will change and for almost free because these books are quite old (but have never been so fashionnable).

• Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977
• Chaos: Making a New Science, by James Gleick (You will learn here the link between fractals and chaos. In two words: fractals can be seen as the domain of convergence of a dynamical system)

Nostalgia: I was offered these books 13 years ago. I read them then. They are definitely among the best gifts I have ever received. They stand in my bookshelf, one arm away from where I am writing these lines. Believe me these books are far easier to read than the Bible so you can give them to your children. I read them again 2 years ago after completing my studies in Dynamical Systems and Controls. I understood plenty of things I had missed in the first reading. So you can also offer these books to adults. They will learn quite a few things about the nature of our world.

Resources

In English

• Now is the time for a revolution in economic thought, Anatole Kaletsky, Times 9 February 2009
• In Plato’s cave, The Economist, 22 January 2009 (a very interesting article which mentions Mandelbrot)
• The Risk Maverick, video with Nassim Nicholas Taleb and Benoît Mandelbrot, 21 October 2008 and an excellent summary of this video
• A focus on the exceptions that prove the rule, Benoît Mandelbrot & Nassim Nicholas Taleb, Financial Times, 23 March 2006
• How Fractals Can Explain What’s Wrong with Wall Street, Benoît Mandelbrot, Scientific American, February 1999
• Mandelbrot’s publications on finance
• Benoit Mandelbrot, bio in english
• The Fractal Geometry of Nature, by Benoît Mandelbrot; W H Freeman & Co, 1982
• Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977
• Chaos: Making a New Science, by James Gleick

In French

• Redécouvrir Mandelbrot, Philippe Herlin, La Tribune 5 février 2009
• Les fondements erronés de la finance, Philippe Herlin, Les Echos, 5 Janvier 2009
• Philippe Herlin
• Benoît Mandelbrot, bio en français

If you know other good links about fractal, chaos theory and economy, please leave them in the comments.