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andi999

Here is the insane thing about pi: any digit of pi can be calculated directly in hexadecimal.

https://en.m.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80...

Like, you can compute digit 10billion without knowing any of the previous digits (in hex..)

If this doesnt prove the matrix then I do not know what does ;-)

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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)

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.

And, of course, Fabrice Bellard found a faster version of this formula in his copious spare time: https://en.m.wikipedia.org/wiki/Bellard%27s_formula

In between making tv transmitters with video cards, winning ioccc’s, booting Linux in browsers, ...

The guy is a mad man. I pity his keyboard.

How utterly strange. Thanks for that.

Quite. And, it has me wondering.

Would that not mean that we then have just as quick of a way to calculate an arbitrary decimal digit of pi? Hexa and dec feel geometrically close enough for any dec digit to be fully "covered" by a couple calculations of contiguous hexa digits.

From what I've read, it's not equally easy, and nobody has figured out a way yet, including the really smart people who came up with the hex algorithm.

Maybe there is something special about a power of two base.

How deliciously intriguing. I'll have to take some time and try to sink my teeth into this one.. someday :)

e: Although alas, after an ounce of consideration of some blown up exponents for 10 and 16, it feels a little(read: much more) daunting to find a clean, mechanical conversion.

Does this exemplify that there can be unique properties of a base?

> any digit of pi can be calculated directly in hexadecimal

Curious what you mean by "directly". How many FLOPs does it take to compute the n-th hexadecimal?

Not many! See https://giordano.github.io/blog/2017-11-21-hexadecimal-pi/

If I'm reading that right, it's O(n) to compute the n-th hexit.

so that isn't really directly. Directly is O(1)

Directly doesn't mean immediately here. It means without computing intervening digits.

What is it about hexadecial that makes this happen?

How do you verify if it’s correct?

Once you guess the formula, the proof is just a few lines of calculus. It's accessible to anyone who has taken Calc I, I think. The full text .pdf of the original paper [0] is freely available; the result is Theorem 1, whose proof is on pages 2 and 3.

I believe the formula was found by computer search, but my memory could be failing me. The three people the result is named after are all well-known for computer-assisted mathematics, for instance using the PSLQ algorithm [1].

[0]: https://www.ams.org/journals/mcom/1997-66-218/S0025-5718-97-...

[1]: https://en.wikipedia.org/wiki/Integer_relation_algorithm

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