Delving into 762 decimal places of pi, a remarkable row of nines occurs.
Six nines in pi - Wikipedia
If we were to round off those nines and construct a circle with an inaccuracy/error size of one Planck Length, how big would the circle be?
Larger than the Observable Universe?
(IIRC, there are 65 orders of magnitude between Planck and Cosmic scales)
What does this have to do with Feynman??
Anyway, the circumference of a circle is proportional to its radius. The usual way I have heard this problem is, suppose you have a belt snugly fitting around the Earth (about 40,000 km), and you lengthen it by 1m. Drawing it again into a tight circle, how high off the Earth’s surface will it be? [Answer: 16 cm]
Planck’s length is around, let’s call it roughly 10−35 m. So no need to even approach 762 digits to get that uncertainty. Even if you throw in the radius of the observable universe which is only another 27 orders of magnitude or so.
I can’t speak for the OP but I think this is where Feynman comes into it:
This sequence of six nines is sometimes called the “Feynman point”,[5] after physicist Richard Feynman, who allegedly stated this same idea in a lecture.[6] However it is not clear when, or even if, Feynman made such a statement. It is not mentioned in published biographies or in his autobiographies, and is unknown to his biographer, James Gleick.[7] - SOURCE (same link as in the OP)
Didn’t you answer the question in your own OP? 762 digits of pi, 65 orders of magnitude between Planck and the cosmos. 762 is greater than 65.
Yes, but I am a flawed human being prone to error.
Interesting. According to wiki, the first to actually compute the digits in this range was an amateur named Ferguson in the 1940s, using an electromechanical desk calculator. I wonder if he briefly thought he’d hit convergence.
Kind of hard to compute digits using a series that does not converge.
Yeah, wrong term. What did I mean? Finiteness?
Proofs that pi is irrational have been known for centuries. We can probably assume that Mr Ferguson, having started to calculate pi, was aware of this irrationality. If the three nines gave him any idea that he might have reached the bottom of the well, so to speak, it would have lasted only very briefly.
Very. But not impossible. IIRC, some of Ramanujan’s methods used non-converging series.