The Buffon's needle experiment is one that you can repeat many times, and in the limit it will give you an increasingly accurate estimate of pi the more you repeat the experiment.
Simply stated: you throw a knitting needle on a floor, with wooden floorboards, many times and you count how many times the needle crosses any line where two floorboards join. The probability of crossing a set of parallel lines is rather elegantly related to the number pi.
Now, you can of course do this physical experiment as a computer simulation to generate pi. But for the simulation you need random seeds to generate random positions of the needle with respect to the parallel lines.
The question is, can you start with a first few thousand known digits of pi as random seeds, in the simulation, and generate the rest of pi "out of itself" ?
Even though this is a number system, and not a physical system, it seems to me this problem is intimately related to the Second Law of Thermodynamics and the answer ought to be no: we cannot generate pi out of itself.
But the question is, how do we go about formally proving this? Any ideas?
If you can prove your statement and elucidate a connection to thermodynamics then you have answered the question.
Derek, unless you have an infinity of throws, you cannot have infinite accuracy, and I'm pretty sure you can demonstrate that even with an infinity of throws there would still exist residual uncertainty.
This being said, I understand that your question would relate to generating a large but finite series of numbers in the pi sequence, so you don't necessarily have the infinity issue. Somehow however this self-generation does not feel right, I'm sure your hunch is correct: no can do. Let me think about how to formally prove this, I sense that there is probably a way to prove this by 'reductio ad absurdum'.
Of course, if the inaccurate digits you get when you don't have enough statistics are random, then you can use the incorrect value of pi to generate the correct value of pi.
The value of Pi is related to the probability P of crossing. However, during the simulation, this probability can only be approximated as a ratio of two integers. Consequently, any estimation of Pi that would be deduced from this approximate probability would not yield a series of truly random digits and would therefore introduce a bias in the simulation.
One of the problems with generating pi out of itself is that that the sequence of numbers that one is attempting to use as a random seed won't be random enough if there is any correlation.
So how about this: we run two simulations in parallel. One is a simulation to generate Euler's constant out of random seeds, and the other generates pi out of random seeds. You feed in the output digits of one simulation to act as the seed for the other simulation.
By alternating between these two simulations, in this way, you could possibly break up any correlation.
I wonder if that could work?
Thinking about this again, I think in this case that pi is contained entirely in the problem description itself. That is to say, since Buffon's needle in the limit will produce pi almost surely, the measurements themselves must not contain any information on pi. In principle it should therefore be possible to make something like this work.
Perhaps the best counterexamples, though, are the series representations of pi, which function without any randomness at all. But that's fine, because pi isn't random.
Returning to the original question involving Buffon's needle, I have attempted to do this, and so far have not been successful. Depending upon how I extract randomness from the current estimate of pi, the estimate either gets stuck or diverges. However, Buffon's needle does not depend upon the randomness of the position that the needle lands in, but just on its distribution. As long as you choose your method of randomness extraction in such a way that it does not deviate too much from a uniform distribution over the short term, you will get the correct value of pi.
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