Chaos game: the random selection of points can get the order


What if we took a starting point whatever, we move to a new point corresponding to half the distance against some fixed points (and I repeat it for a sufficient number of times)?

It happens, for example, that shown in the following image.


The space between the 4 points [(1, 1), (9, 1), (5, 9), (5, -7)] is “uniformly” occupied by new points chosen as before said.

And up there, nothing particularly interesting.  Which is why you do not understand what for, in the video The Strange New Science of Chaos,  it starts from there.



Here the figure of the game of chaos, made in the traditional way:

  • a dice to choose at what point move
  • only 3 points (which correspond to the faces of the dice)



After a little Python programming (if you are interested you ask for the code via the contact form) and a few seconds to run, here is the incredible figure emerging from the use of only three points [(1, 1), ( 9, 1), (5, 9)] and 30 000 steps from the starting point of coordinates (2, 2).



From that comes the order in the figure that looks like a fractal?


By repeating the simulation many times, you can verify that it is not true that only in case of the three points “attractors”  pattern occurs.

See the following picture: there are 6 “attractors” (of which only 5 visible, because one covered by other points, calculated at random, of the game).
The ‘attractors’ are positioned in
[(5.77, 1.77]), (5.15, 0.080), (9.59, 0450), (7753, 5114), (0468, 1714), (2761, 9067) ]

The starting point has value (2,286, 2,709).


As we see, in the left, where is the prevailing the side formed by the two point (0468, 1714) and (2761, 9067), there’s a triangular pattern, revealing some triangles in the opposite direction to the side, as in the case of only 3 “attractors”, precisely.

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