Talk:Pi

The article says:

    It was shown that the existence of the above mentioned Bailey-Borwein-Plouffe 
    formula and similar formulas imply that the normality in base 2 of <eth> and various 
    other constants can be reduced to a plausible conjecture of chaos theory. 

What is the conjecture? Who showed this reduction? Can you give a reference for the papers involved? Or their websites? Even a few keywords suitable for web searching would be very much appreciated!

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I see you've added "Bailey and Crandal in 2000". That's great. It would be good to add a link to their paper (or it's abstract) if it exists somewhere on the web. I've tried searching for it, and haven't had any luck.

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But the article does give an URL, and you can find the paper there, in both PDF and PostScript. (The PDF is very badly done, however.) --Zundark, 2001-08-21

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You're right. My mistake. I'd missed the URL in the earlier section.

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Of particular interest to me is this: if we live in a non Euclidian universe, does that alter the value of pi? Is it possible that a non euclidian universe would render pi a distance dependedt function? Just musing, really.

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I define Pi as a function of the distance metric in a metric space: Pi equals half the arc length of the curve created by the locus of points of distance 1 from a given point. In a hexagonal world such as that used in many turn-based video games, Pi == 3. In a geometry with distance metric d((x1, y1), (x2, y2)) == (abs(x2 - x1) + abs(y2 - y1)) such as city blocks, Pi == 4. Of course, the familiar Euclidean distance metric provides a value of Pi just a bit more than 355/113, and nearly all digital signal processing takes place in Euclidean geometry.

In geometries that don't preserve lengths of translated lines (such as the geometry of curved spacetime), "distance 1" is meaningless, and Pi depends on the location and the radius.

-- Damian Yerrick

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That is not the standard definition however: Pi is a well defined real number, and it has nothing to do with geometry. It is always 3.14.. no matter what. Mathematical constants don't depend on physical contingencies. If our world is not Euclidean, then there will be some circles of diameter one whose circumference is different from Pi. --AxelBoldt

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I deleted the image at http://www.nersc.gov/~dhbailey/dhb-form.gif from the article. It might be OK to swipe the image and put it on the Wikipedia server, but it's not OK to link to the image itself on the government server. We don't own their bandwidth... --LMS

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I've got no problem with showing a particularly long value of pi, assuming it's accurate as far as it goes. It may be curious and totally useless, but it's not vandalism. Eclecticology

• I called it vandalism because it's merely part of a days long pattern from this character. Rgamble

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Geometry was a purely axiomatic mathematic until Cartesian thought entered. You can't claim that these equations are Euclidean, Euclid wouldn't have recognized them.

They are certainly not from "plane geometry" since some of them talk about three dimensional objects. The "Euclidean" was there to emphasize that there are other geometries where these formulas are wrong. But that point was made earlier already, so I guess we can just call it "geometry". AxelBoldt 03:08 Oct 1, 2002 (UTC)

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Large value of pi question: Is this true?

pi to 40 places is sufficient to measure the circumference of the known universe with an error less than the width of a hydrogen atom.

If so, what is the purpose of calculating pi to large numbers of digits?Ortolan88

What is the purpose of climbing the Everest? It is there. Same with pi.--AN

But is the statement true? If so, it, or some variant, should be in the article to give some proportion to this quest. I would also like to know what is so hard about calculating that number of digits. Is it anything more than long, long, long division? In what way does this advance the art of mathematics? These are serious questions by an uninformed member of the encyclopedia-reading public. Ortolan88

Computing many digits of pi is trivial and not mathematically interesting. It is often done to test (super-)computer hardware. I don't know if the statement about the universe is true, but something close to it is probably correct. You don't need more than a couple dozen digits in real word applications. AxelBoldt 21:07 Nov 23, 2002 (UTC)

According to the article on pi, "The most pressing open question about π is whether it is normal, i.e. whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely randomly." This can only be answered by a mathematical study of the digit sequence. Obviously one wouldn't study it by looking it up in an encyclopaedia, but even so, people at home might want to do a bit of a check for themselves, to see if it sounds plausible. :) -- Oliver Pereira 00:20 Nov 24, 2002 (UTC)

Evidence for or against the normality of pi can obviously never be produced by looking at a finite initial segment of the digit sequence. That's why I said above that computing digits of pi is mathematically uninteresting.

But people should definitely be encouraged to experiment with the digits. So we should tell them how to generate the digits for themselves, so that they have them in a format which allows all sorts of statistical tests. Right now, using our data to find out how often the digit sequence "33" occurs among the first 10,000 digits would require a perl program that's not much simpler than directly generating the digits from scratch. AxelBoldt 20:42 Nov 24, 2002 (UTC)

Yep, I agree - except about the simplicity of writing a perl script to compute the value of pi! :) Since we are writing an encyclopaedia for general use, most people referring to it will just be ordinary people without any programming ability. Some of whom genuinely believe that pi is exactly 22/7! Such people might want to see a concrete demonstration of just how random the digits of pi are. Plotting bar charts of the frequencies of digits might be an amusing mini-project for a schoolchild who is starting to learn about statistics. Well, maybe not, but I expect there are many more plausible suggestions that people could come up with. I just mean that members of the public might be curious to see the digits of pi and play with them, even if they don't have the ability to do anything serious with them. I'm thinking of a past version of me when I was at school, for example. -- Oliver Pereira 21:38 Nov 24, 2002 (UTC)

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I removed this:

It is said that pi to 40 places is sufficient to measure the circumference of the known universe with an error less than the width of a hydrogen atom.

Who says that? If somebody has done the calculation, we can simply omit the qualifying "it is said that" which essentially renders the whole paragraph pointless. AxelBoldt 20:00 Nov 25, 2002 (UTC)

I'm confused about it myself. Surely the accuracy of the result is dependent on the accuracy of the figure we have for the diameter of the universe, as well as the accuracy of pi. -- Tarquin 20:07 Nov 25, 2002 (UTC)

It isn't pointless. It gives a sense of proportion to the whole thing. 22/7 is sufficiently accurate for pi to get the circumference of a can of corn. With a circle of one mile diameter 3.14 yields 16572.2 feet circumferences, 3.142 gives 16589.76, 3.1416 gives 16587.648, and 3.14159 yields 16587.595, which is where my calculator poops out, but, obviously, the more digits of pi are adding accuracy. If one of you cosmo brains would do the same arithmetic for some galactic measure or other (instead of silently deleting my interesting, if unproven statement about hydrogen molecules) using, say 40 places for pi, then maybe we could put this million digits of pi business into some kind of proportion. What is confusing or pointless about that? This is an encyclopedia. My whole intent is to give readers some idea of the value of the additional digits of pi.Ortolan88

Okay, i'll have a go... for a universe 12 billion light-years across, that's about 1.135e26 m. An atom of hydrogen is about 1e-10 m across, meaning you'd need about 36 decimal places in pi to get error levels below the diameter of hydrogen. So 40 does it admirably. Graft

The observable universe currently has a radius of about 50 billion lightyears, because of the past expansion, but your estimates still work. Of course, if you want the volume of the observable universe precise to the volume of a helium atom, you need about 270 digits of pi. AxelBoldt 23:17 Nov 30, 2002 (UTC)

Cool, so I assume you'll add this to the article? Ortolan88 05:57 Nov 29, 2002 (UTC)

PS - OK, no takers, some time this weekend I'll add it to the article. It isn't just whim, folks, it is something the average encyclopedia reader deserves and will only get from the Wikipedia. I may move one of the more interesting paragraphs up a bit too. And, mathematicians, I read in a rival encyclopedia [1] (http://www.encyclopedia.com/html/p1/pi.asp), that Euler came up with a connection between pi and natural logarithms that isn't mentioned in our article. Ortolan88 15:40 Dec 7, 2002 (UTC)

This just in, pi to 1.24 trillion digits. [2] (http://seattlepi.nwsource.com/national/98912_pi07.shtml) How about an article listing them all? Ortolan88 22:12 Dec 7, 2002 (UTC)

Thats goign to take more than 1 Terabyte of memeory!!! :-S

Regarading the rival encyclopedia article: it says "the famous formula ei 1, where i 1 ." now unless that's my browser skipping characters, they've made a complete pig's ear of Euler's identity... oh yes, by the way... we have a full article on that equation. nananana! ;-) -- Tarquin 22:22 Dec 7, 2002 (UTC)

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These approximations were once useful to the applied sciences; the more recent approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers.

I don't understand the above sentence.

• Of what use were "these approximations"?
• Why aren't they useful any more?
• Are the recent approximations entirely useless, or just the extra digits?

Without answers to all the above questions, I'm inclined to delete the quoted sentence. But I hope someone who knows a lot more about math than I do (like Axelboldt) can help out here. --Ed Poor

it means this: newer approximations have more digits. They don't make the old ones obsolete. An old approximation of, say 50 digits is useful. A 10,000 digit is pointless; the extra accuracy is irrelevant because we never need it. It's hust the extra digits that are irrelevant. The 50th digit (for example) has not changed since it was first determined -- that's something for an article on approximations[?] to explain: it is normally possible to know how good an approximation is: we don't just calculate 50 digits, we know that those 50 digits are right, and even if we went further, we'd still get them. hope that helps -- Tarquin

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This entire debate is patently ridiculous and should be deleted. The children with their pocket calculators should go off and let the rest of us work in peace.

First, using pi calculations to calibrate supercomputer speeds is absolutely meaningless. Supercomputers are meant for real work, not moronic games. Calibrate speed with real work.

I don't know where you got the idea that pi calculations are used to "calibrate supercomputer speeds" nor do I know what that even means. Pi calculations (and prime number calculations) are often used to test new supercomputers. You let them compute digits of pi for a week with two different methods and then compare the results, to check for hardware bugs. AxelBoldt 05:13 Nov 26, 2002 (UTC)

Second, nobody needs to know pi to more than five digits. Period. End of discussion. Anything more than that is simply compensation. Something so delicate as to need outrageous values of pi is a toy, that's all.

Third, every resource spent analysing pi is a resource wasted. Spending time on pi is exactly as fruitful as spending time with the human genome.

Some people think the human genome will someday in the future produce a useful medicine or treatment. Some people also think that pi may have a similar surprising usage. Psi 12:37 Dec 2, 2002 (UTC)

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I am inclined to question the inclusion of the physics formulae. Surely the appearance of pi in these is simply a quirk of the definition of the physical constants such as Plank's constant and the gravitational constant. The significance of a physical constant tends to be recognised early in the development of the theory in which it features. When different derivations are made from the theory sometimes a factor of pi will appear in a formula, and sometimes not, for detailed mathematical reasons (eg the inversion of a fourier transform).

But I believe no matter how you redefine the physical constants, pi will always show up in your fundamental equations, just in different locations. For instance, if you redefine G to get rid of pi in Einstein's equation, it will then show up in Newton's law of Gravity. So we may as well list the locations that pi shows up in our accepted system of physical constants. AxelBoldt 23:17 Nov 30, 2002 (UTC)

This is true. The same issue arises in electromagnetism and is further exacerbated by the contrast between MKS and CGS units. In the Coulomb force law for electrostatics (at least in SI units) Pi does appear, which, in turn, results in its not appearing in Maxwell's equations. In quantum theory an explicit reference to Pi can be made to appear or disappear from an equation by changing from h to h-bar. Perhaps this is worth explaining in the article.

If we want a sample equation from physics where the appearance of Pi is less arbitrary then I think the period of small oscillations of a pendulum in a uniform gravitational field would be a better candidate. -- Alan Peakall 12:25 Dec 2, 2002 (UTC)

Also I believe the mnemonic linked to Isaac Asimov was coined by James Jeans -- Alan Peakall 12:00 Nov 29, 2002 (UTC)

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In a long ago and fruitless sojourn into the land of entry-level statistics, I seem to remember that statisticians use a wholly different pi that stands for some variable or another. The statistical use probably doesn't deserve a whole article, but there should be a mention that the same Greek letter is used in statistics too, if anybody knows exactly what it stands for. Tokerboy 23:03 Dec 7, 2002 (UTC)

It sounds vaguely familiar. There's ddefinitely pi bonds[?] in chemistry. -- Tarquin

Oh they use the symbol pi to represent profit in economics. -- Mark Ryan

See pi (letter) for various usages of the Greek letter in different fields. SCCarlson