new chestnut: pi as a pattern

Let's say we have a very simple system that can be in any one of ten discrete states. These states we number from one to nine. There are rules for how this system changes from state to state, but we have to figure them out for ourselves, just from observing the system's state over time. Now, we can see patterns in the way systems like this behave. Look at this one:

1,2,3,3,3,3,3,3,3,3,3,...

9,8,7,6,5,4,3,3,3,3,3,...

We can be fairly sure that this one settles on 3, no matter where it starts. Once it reaches 3, it stops being interesting because it is in a stable state. We know exactly what it is going to do from then on.

This one:

1,3,5,7,9,1,3,5,7,9,1,3,5,7,9,...

Cycles through odd numbers. Again, it is in a stable state. A more complex one -- a cycle -- but it's predictable.

But what if the system gave us this output:

3,1,4,1,5,9,2,6,5,...

Can we say that this is a pattern? Is this system in a "stable state?" You tell me.

Successive digits of pi do form a predictable sequence (otherwise we wouldn't be able to calculate them), but it is not a stable one in the sense of eventually becoming repetitive.

The reasons are simple enough: pi is irrational, as proved here, whereas all decimals which display infinitely repeating patterns are rational. In fact, given a repeating decimal, any high school student can recognise it as representing the sum of a geometric progression and, using the formulas given here, calculate exactly which rational number it corresponds to. Therefore we know that the decimal expansion of pi never reaches a point where some string of digits is repeated forever.