The billiard ball example

Posted by  Shawn Callahan —October 8, 2007
Filed in Fun

When I talk about complexity to clients I mention that complex systems are impossible to predict in detail especially as your forcast extends into the future. I point out that there are so many connections among the objects affecting the system and many of the cause and effect relationships are non-linear (a small thing can have a big impact and vice versa). Every now and then someone will say, “but if you could work out all those connections you could predict the outcome.” And this is where I will tell them the chessboard story.

The legendary information scientist, Claude Shannon, calculated how many possible moves there are on a chessboard. It’s a finite system of 64 squares, 32 pieces, 6 movement patterns. The number is big and equates to the number of milliseconds the world has been in existence. And that’s for a simple system. Imagine the possibilities in a social system where the objects have free will.

But I think I’ve just read a better analogy (perhaps equally as impressive) and the topic is billiards. The calculations were done by Prof. Sir Michael Berry in 1978 in his paper Regular and Irregular Motion, in Nonlinear Mechanics and recounted in The Black Swan.

If you know a set of basic parameters concerning the ball at rest, can computer the resistance of the table (quite elementary), and can gauge the strength of the impact, then it is rather easy to predict what would happen at the first hit. The second impact becomes more complicated, but possible; and more precision is called for. The problem is that to correctly computer the ninth impact, you need to take account the gravitational pull of someone standing next to the table (modestly, Berry’s computations use a weight of less than 150 pounds). And to compute the fifty-sixth impact, every single elementary particle in the universe needs to be present in your assumptions! An electron at the edge of the universe, separated from us by 10 billion light-years, must figure in the calculations, since it exerts a meaningful effect on the outcome. (p. 178)

No wonder I can’t play billiards to save myself.

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About  Shawn Callahan

Shawn, author of Putting Stories to Work, is one of the world's leading business storytelling consultants. He helps executive teams find and tell the story of their strategy. When he is not working on strategy communication, Shawn is helping leaders find and tell business stories to engage, to influence and to inspire. Shawn works with Global 1000 companies including Shell, IBM, SAP, Bayer, Microsoft & Danone. Connect with Shawn on:


  1. Thanks for that one, I may refer others to it. I’m constantly running into arguments with people who believe complexity and chaos are synonymous with ignorance – a smokescreen for “I don’t know” – and an excuse to settle for alternative simplistic beliefs.
    Do you know, are the Berry calculations real ? Or simply illustrative ? (Given the paper is 1978, I’m amazed I’ve not come across a reference before – you live and learn.)

  2. H Ian, I’m really relying on Nassim Taleb’s description because I can’t make head nor tail of the Berry paper – way too mathematical for me. So not really sure what you mean by asking whether the calculations are real but my impression is that Berry did make the calculations and they are quite complicated.

  3. Ningmeng says:

    The part of Berry’s paper that directly refers to billiards (p99) is quite easy to understand. The rest, which also talks about chaos, is not necessary to understand how he go to these numbers.

  4. Ivin says:

    The explanation of this example is found starting on page 95 of the paper. The paper was scanned in the wrong order, so the explanation runs from pg 94 to 96, then back to 95, and finally to 97. It written in terms of a “hard sphered fluid”, but Nassim Taleb has correctly characterized the result. It is essentially that to achieve a degree of accuracy within one radian in predicting the point of impact and angle of reflection, at 56 collisions, you would need to predict electrons at “the limit of the observable universe, i.e. D~10^10 light years” away.

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