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Comment by drewolbrich | Hacker News Reader

If you know the diameter of the observable Universe and you want to calculate its circumference with the accuracy of the diameter of a proton, the number of digits of pi that you need is 43.

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The smallest possible distance is the Plank lenght, 1,610^-35 (110^-15 is the diameter of a proton). And for that you only need around 60 digits of pi to calculate the circumference of the universe.

Of course, that is just for the simple operation of calculate the circumference given the diameter, more complex operations with pi may require more precision.

Never heard it stated that way. Can you elaborate on "the smallest possible distance is Plank's length"? Is that the smallest observable distance?

> According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.

At least according to Wikipedia, it seems it is indeed the smallest observable distance. Although it has never been proven and follows from the theoretical generalized uncertainty principle. https://en.m.wikipedia.org/wiki/Planck_length#Theoretical_si...

If you have some time to spare, watch this talk "The astonishing simplicity of everything" by Neil Turok. He explains all this so beautifully.

https://www.youtube.com/watch?v=f1x9lgX8GaE

Well presented and really gets you thirsty for more, excellent talk – thank you very much for sharing!

He was painful to watch, I gave up after a while. Maybe it's just me.

http://physics.stackexchange.com/questions/28720/how-to-get-...

From what I understand the particles you need to probe distances on the order of the Planck length are so energetic that their own gravity would start interfering. Their Swarzschild radius would become bigger than the distance you're trying to measure.

Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nature of mathematics? Seems like you could argue that digits beyond what are needed to render measurement to within one Planck length are meaningless and therefore a kind of fiction... at least if you take that philosophical position.

Pi is far more than just the ratio of the diameter to the circumference, even as a constructivist: https://affinemess.quora.com/What-is-math-pi-math-and-while-...

It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the precision of numbers involved could also keep growing.

If the universe were a computer, it would not have to compute π to "simulate" physical processes. Mechanical processes don't depend on the value of π directly.

If universe had to know every constant somehow involved in the process, it would not be able to simulate computers, because it involves uncomputable numbers: https://en.wikipedia.org/wiki/Chaitin%27s_constant#Uncomputa...

And wave functions collapse only upon measurement...clever lazy initialization

Pi has other uses. The digits are real, but they just don't matter in practical engineering.

But lets assume that there really are 10 dimensions - in that case a volume of a 10 dimensional sphere will require (pi^5)*r^10.

If you want to measure the volume to 1 plank 10 dimensional cube, you will need more digits.

Pi appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc.

There is a list of formulae where π appears (some related to circles, some not) here: https://en.wikipedia.org/wiki/List_of_formulae_involving_π

But how many coins can you flip before the heat death of the universe, and does using 43 digits make your expectation off by more than say 1/1000000?

In other words, if you were trying to calculate pi by checking the actual probabilities, how many digits can you get?

But pi is not an empirical constant. It is a mathematical constant, with well-defined formulas such as pi/4 = 1-1/3+1/5-1/7+...

Would have been so much better if that answer was 42.

It was 42, but in the time it came to get that answer, it kept expanding. So we need a new computer.

Gonna have to think deeply about the design of this new computer. ;)

Well, reality can be inaccurate at times.

I've always heard the number is around 50, but do you have any references to show that as being correct?

This [1] puts it at 44.

[1] http://www.wolframalpha.com/input/?i=observable+universe+dia...

I think this assumes the order of magnitude of the size of the observable universe.

Clearly if the diameter of the universe is smaller than a proton then you don't need 43 digits of pi to calculate its circumference to smaller than the diameter of a proton. So if the diameter of the universe is 10^1000000^100000000 proton wide the precision you need for pi would be way higher than 43?

It does depend on the size of the observable universe, which we do know because, well, it's observable.

The post i replied to started off with "If you know the diameter of the observable Universe" but if the result it depends on the size of the observable universe then you ought to say "given the size of the observable universe".

Starting with 'if' would imply that the result only requires you 'knowing' the size of the obsv. unv. but does not depend on its actual size.

I think the meaning of "If you know," in this case was more like, "If you take as an input."

Yep. "If you know X" is commonly used as "given X".

There's a difference between "the size of the universe" and "the size of the observable universe", see also "cosmic microwave background", "dark energy", and "inflation."

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