4 ms·

While this is true, the fact remains that it gets harder and harder to calculate (as far as we know). You can see in the given formula that the base-16 multipli

While this is true, the fact remains that it gets harder and harder to calculate (as far as we know). You can see in the given formula that the base-16 multiplicand is not an integer, so it "bleeds" into lower digits and consequently also receives interference from higher digits (which themselves also have interference, and so on, although the decision converges much sooner than the origin). There are analogies of digits of prime to sort of infinite libraries, kind of an evolving random universe of numbers. But the digits of pi are not a local (fixed-size) function of the previous digits, so the analogy to a physical simulation (or cellular automaton) isn't quite right. But the idea of a cellular automaton with varying locality (in this case increasing radius of interaction) is itself quite interesting to me. (in this case it's a logarithmic neighborhood and the rule is chaotic for the initial conditions)

gnramires 6y ago
Now that I think about it, it should be obvious the size of dependency cannot be fixed, otherwise pi would be periodic! (there are finitely many fixed size 'parents', so it must recur) I should also note that the straightforward interpretation in this case is that of a temporal neighborhood for a Cellular Automaton! That is, dependence of several states back in time, and 0 space dimensions. You can also think of a 1D CA if you introduce a special state that signals the "expansion" of the digits of pi (which digit we're currently expanding) This also enlightens me in the bizarre concept of multiple time dimensions. If you start with a 2D field, and use the same technique of keeping track of the current active expanding cells (i.e. "current time"), starting from a single active cell (time 0) in a top-left corner, then you can expand cells across a diagonal, and they depend on previous states in two different directions.

gnramires 6y ago

Now that I think about it, it should be obvious the size of dependency cannot be fixed, otherwise pi would be periodic! (there are finitely many fixed size 'parents', so it must recur) I should also note that the straightforward interpretation in this case is that of a temporal neighborhood for a Cellular Automaton! That is, dependence of several states back in time, and 0 space dimensions. You can also think of a 1D CA if you introduce a special state that signals the "expansion" of the digits of pi (which digit we're currently expanding) This also enlightens me in the bizarre concept of multiple time dimensions. If you start with a 2D field, and use the same technique of keeping track of the current active expanding cells (i.e. "current time"), starting from a single active cell (time 0) in a top-left corner, then you can expand cells across a diagonal, and they depend on previous states in two different directions.