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But those of system (d) appear at positions {3, 15, 568} and those of (e) at {855, 4} .
It is rather common for the active node eventually to get stuck at a particular position in the network; the picture below shows the effect of this on the total number of nodes in the last case illustrated above.
Multiplication by 2 turns out to correspond just to shifting all digits in base 2 one position to the left, so that the overall pattern produced in this case is very simple.
being able to see a sequence of explicit bumps in the road, the initial conditions for the position of a point in the kneading process are encoded in a more abstract form as a sequence of digits.
But if one progressively changes the initial position of the ball, then when one passes the center a discrete transition occurs, and the ball instead rolls into the right-hand minimum.
If one managed to find the positions and properties of all the particles in the system, then no information about the system would remain unknown, and the entropy of the system would just be zero.
Present-day physics almost always assumes that space is a perfect continuum, in which objects can be placed at absolutely any position.
And while the positions of particular updating events are different in every picture, the point is that the network of causal connections between these events is always exactly the same.
But just as in the case of position there is no contradiction with sufficiently small underlying discrete elements.
For it typically will not matter much which of the standard methods of perception and analysis we use: after trying a few of them we will almost always be in a position to come to a quite definite conclusion about whether or not something should be considered random.