After reading the “Who invented Pi” mailbag item, it jogged my memory of something I learned in an advanced statistics class (many moons ago). But first, the question:

Is there a natural derivation of Pi that has nothing to do with circles, diameters, circumferences, or spheres?

I seem to recall a statistics question that related to the probability of dropping a stick (line segment, unit length 1) randomly above a square (unit length and width of 1), and the answer was something like Pi/4.

So, was I simply consuming too many drugs at the time, or is Pi a natural occuring phenomenon, even without regards to circles? Or is there an implication of a circle from the scenario presented above that is simply eluding me.

Search Google for “Buffon’s needles.” I read recently that although Buffon’s method is valid, it is likely that he cheated, because his answer is too close given the number of trials he used.

Pi is of very frequent occurrence in statistics. It is, for example, part of the formula for the Gaussian (normal) distribution.

There are very, very many derivations of [symbol]p[/symbol] that have nothing to do with circles. For instance, the series whose nth term is 1/n[sup]2[/sup] is equal to [symbol]p[/symbol][sup]2[/sup]/6. There is an algorithm for explicitly calculating the nth hexadecimal digit of [symbol]p[/symbol].

I have absolutely no business being in a math thread, but what the hey.

IIRC, there are a few folks who hold the position that the ratio of a river’s length (source to mouth:as the crow flies) is just about Pi. The thinking is that every bend in the river increases the rate of flow on the outside of the river. This has to be compensated for with a corresponding bend that turns the river back onto itself. Regardless of how, er, kinky the river is, the ratio is pretty much fixed.

Three things worth noting:
Sure it’s kinda based on circles, but only kinda.
I don’t know if this is what you had in mind when you said natural.
Math people will proabably crucify me for the above.

Looks like Coileán is giving a ratio of the total length of a river, including all the curves and bends, to the total length of the river expressed as a straight line from the beginning to the end, ultrafilter.

I’d think any relation would be coincidental and non-universal, but this is the first time I’ve even heard of the idea. It would also depend on how you define a river.

On the other hand, I’ve seen some rivers where the source is almost next to the mouth, but geography interferes and forces it to take a huge loop around.

In thinking about it some more, I don’t think that it’s actually possible for there to be any direct pattern like that. If it’s true of all rivers, then it ought to be true for any segment of a given river. So, let’s look at a point halfway down the length of a river. If the ratio of length is constant, then the crow-flight distance from head to mouth would need to be equal to the crow-flight distance from head to midpoint, plus the crow-flight distance from midpoint to mouth. But this would only be true if head, midpoint, and mouth were on a straight line, which is not in general the case.

As for other occurances of [symbol]p[/symbol], everything involving [symbol]p[/symbol] comes down to circles in the end, but sometimes it’s a more direct connection than others. Buffon’s Needle Problem, for instance, reduces to the area under the curve of a sine function. But the trignometric functions are all derived from circles. Ultimately, the presence of [symbol]p[/symbol] in your answer means that the problem eventually works its way down to trig functions or exponentials, which are closely related to trig functions.

OK, so my last comment, which was, “Or is there an implication of a circle from the scenario presented above that is simply eluding me” sounds like the real answer.

That is, while there are some natural occurances of p, in the end, it still relates or derives from circles. In the Buffon’s Needle example, it has to do with the fact that the randomizing of the angle of the needle eventually approximates a circle, and that’s how p gets into the picture.

Stated a little differently, since circles occur in nature, p occurs in nature. And in cases where p can be derived without apparent reference to circles, they are in there somewhere. You just have to look.

I would certainly be fascinated by any examples where this would not be true.

Well, [symbol]p[/symbol] is defined in terms of circles, so there’s no way to know that what you’ve got is related to [symbol]p[/symbol] unless you can find a reference to a circle buried somewhere deep within the problem. But sometimes, you do have to dig awfully deep.

I have to say that the article, and this thread, are a little hard to read, because instead of the word “pi” or a character that looks like the greek letter, I see simply the letter p. I’m using the Mozilla browser. So I fired up the page in my secondary browser, Opera, and see that it renders the character as a funny-looking p. Then I tried IE, and see that it shows the greek pi.

So just a note to everyone that unless someone is viewing your writing with IE, they won’t see your fancy symbols.

Yep, and all apologies for the vague wording in the first post. Additionally, having dug up the original reference, it turns out that I didn’t recall correctly.:smack:

The work in question argues that the ratio fluctuates a bit from one river to the next but it tends to hover around Pi. However, if you look at a bunch of rivers, the mean ratio is more or less Pi.

As for the relation being universal, nope. It’s more common in rivers that run through the plains of Brazil and Russia. Coincidental? Dunno. Under the presumption that rivers tend kink in arcs as opposed to angles, it seems resonable to expect that the shape of a circle (semi, quarter, whatever) will pop up. However, I sure as hell don’t think it’s worth arguing over.

Netscape and Mozilla do not render the Symbol font; all others that I know of do. Netscape and Mozilla do, however, support the escape sequences, so π should show up correctly. Let’s see…