My point is to refute the notion that curviness explains irrational length. It does not. The way Cheesic is looking at the world, ALL curves have irrational length, which is not true.
i think some people are confusing the fact that we can’t perfectly measure with the fact that an irrational number can’t be perfectly measured.
an arc is inherently no harder to measure than a straight line, physically speaking. whoever said “yarn” was right.
we approximate, not because it’s curved but because it involves pi. as **chronos ** and **giles **said, there are curves that we can compute without approximations.
Physically speaking, the problem with using yarn to measure is that it stretches.
But physically speaking, we can’t measure anything with infinite precision, which means we can’t tell the difference between a rational and an irrational number by physically measuring.
But this isn’t what Chessic Sense meant by measuring, hence his parenthetical remark “rather, ‘prove the length of.’” In that sense, it is harder to “measure” an arc. In Euclidean geometry, it’s fairy straightforward to prove that one straight line segment is equal in length to another straight line segment. But I think it’s safe to say that, to prove that a curved line segment is equal in length to a straight line segment, or even to define rigorously what we mean by the length of a curve, requires something like limits and, thus, calculus, or at least proto-calculus of the sort Archimedes used.
Where Chessic Sense went wrong was in asserting that this proves the irrationality or incommensurability of a curve (like a circle’s circumference). If this were all it took to prove pi’s irrationality, it wouldn’t have taken until 1761.
But in an isosceles right triangle, the sides are 1, 1, sqrt(2). The latter is irrational and the triangle is clearly a polygon.
Well, yeah. But I was thinking of regular polygons.
Guys, guys, guys. I never said anything about a curve’s length being irrational. I’m talking about ratio of radius to arc length, which we know to involve pi.
Of course you can have rational-length curves. And you can have irrational radii and hypotenuses. And of course a curve can be equal to a line in length. But the thing is, you can’t express a ratio of arc and diameter/radius without using pi. You’re all jumping ahead because you know about pi and you know about irrational numbers, but I started this off by declaring to be behind a “veil of ignorance” about such things.
With no such thing as pi at our disposal and no knowledge of calculus, the only tool we’ve got in our math bags is to use straight lines to approximate the curve. Because those, we know how to measure. If we can just reduce the curve to a line by cutting it up enough, we can just total up all the little lines. Since we still know how to multiply, that’s easy enough to do.
NOW you can apply my explanation. NOW you can see that cutting up a curve just makes baby curves. THAT is why pi goes on forever. And what’s worse is that each baby curve is different than it’s parent! THAT is why pi never repeats itself.
If you ever cut an arc in half and got the same arc, only smaller, then pi would eventually repeat. It’s go 3.141592222222222222222… or whatever and no one would post “what’s the deal with pi?” on the Dope.
Quite the contrary: You’re the one jumping beyond the veil of ignorance, here. Cutting up a cycloid into “baby curves” is also a process that could go on forever, but it gives you a perfectly good rational number (an integer, in fact). On the other hand, when you cut a 45-45-90 triangle in half, you do get the same triangle, only smaller, and yet sqrt(2) is not rational. It’s only the fact that you already know that pi is irrational that leads you to the conclusion you got. But if you didn’t already know that pi was irrational, you couldn’t conclude it from your argument.
Am I the only one who can’t read the thread title without hearing Jerry Seinfeld’s voice? “What’s the deal with pi? I mean come on… is it a number or a dessert? Are we counting or eating? First you count calories, then you eat, then you count the tip. Count, eat, count. Count, eat, count. Newman.”
Note that π is hardly special in this respect. In fact, the digits of π aren’t even known to actually have this property, even though they’re widely suspected to. But it’s easy to come up with other strings of digits which do; for example, “0 1 2 3 4 5 6 7 8 9 00 01 02 03 04 05 06 07 08 09 10 11 12 13 …”, or, for binary, “0 1 00 01 10 11 000 001 010 011 100 101 110 111 0000 …” (spaces inserted for clarity). Every digital file that could ever possibly exist can be found in that sequence at some point, obviously. Of course, when seen to be that mundane a property for a digit-sequence to have, all the fun goes out of it.
This prompted me to see if I could find Firefly S2 in Pi - success! The good news is that it was awesome! The bad news is that those bastards at Pi ran the episodes out of order and cancelled it after only 7 episodes. And I must say, MLK Jr didn’t look nearly as good in River’s revealing outfits as you might expect.
Since there is a distance which is the absolute smallest “pixel size” of the universe (called the Planck length or something, right?) why can’t we get a definite ratio of integers for pi by just measuring the circumference and diameter of a circle using the Planck length as our unit of measurement?
First of all, because we’re talking about circles, not about objects in the Universe. If the Universe does have a pixel size, then that just means that there are no actual circles in the Universe. Second, though, we don’t even know that the Universe does have a limiting “pixel size”: The assumption that it does is just guesses layered on top of guesses supported by yet more guesses.
Unrelated to anything: the ratio of a circle’s circumference to its diameter is usually not really the constant of fundamental interest. The ratio of a circle’s circumference to its radius is much more ubiquitous throughout mathematics. That is, our cultural fixation on π is “wrong”; what we call “2π” is the entity that really matters and deserves a nice name (some propose τ (“tau”); unlikely to catch on, but might as well encourage it as there seem no significant alternative proposals as the moment).
What Chronos said, plus …
There are formulae for calculating π, and they’ve been used to find its value out to billions (trillions?) of decimal places. That right there gives us a ratio of integers far more precise than any measurements we could make of a physical circle.
You obviously have a dream.
Damnit! I wanted to say this! 
I actually came upon this idea completely independently from whoever in mathematics might have realized it first. I stumbled upon it after realizing the equations involving pi tend to have a 2 in them, and the ones that don’t can easily be rationalized. The major two are the area of a circle (A = pi * r^2) and Euler’s Formula (e^(pi * i) = -1).
In the first case, if you draw a radius in and try to calculate the area of the circle like it was a triangle with “base” equal to the circumference of the circle and “height” equal to the radius, you get A = (1/2) * C * r. It’s only expressing C = 2 * pi * r that we get the formula that we usually use that has pi but no 2. To see this slightly better, imagine cutting the circle at the radius and collapsing the circumference down to an infinitesimal distance where the curvature becomes negligible. Not mathematically rigorous, but a good reason to believe the formula.
In the second case, it’s practically just as “mind-blowing” to say e^((2 * pi) * i) = 1. You do get slightly more information in the typical form, as taking the square root of the latter expression does not lead to a definite value of the right hand side. However, taking e to an imaginary power that includes a transcendental and getting an integer out is highly remarkable; any multiple of pi works to get an interesting result.
The only other formula I can think of off-hand that includes a mere pi is the natural log of negative numbers using complex numbers, but log(-1) = pi * i falls straight out of Euler’s formula. If you are taking complex logs, log(1) = 2 * pi * i (* n) anyway; it’s only Log(1) that equals 0.
I encourage anyone else that’s even slightly mathematically inclined that has not thought about this topic before to find as many formulas as they can that involve pi and see just how many have pi as some multiple of 2, and see that those that don’t might make more sense if a 2 was there. Finally, consider that effectively the only formula that uses the diameter of a circle rather than the radius is the one that defines pi. The fundamental mathematical quantity is the radius, not the diameter. It’s only primitive mathematicians that were concerned about the diameter, as it’s far easier to measure for an already existing circle.
I have thought about this as well but speaking as an EE, I’d rather have to deal with the occasional 2π than τ/2. π all by its lonesome appears often, e.g. area of a circle, volume of a sphere, surface area of a sphere, Euler’s identity, etc.
Bytegeist, it appears that the record at the moment is that the first 5 trillion digits of π have been calculated and that the quadrillionth bit of π has been calculated (but without calculating all the previous bits):
Euler’s identity has nothing fundamentally to do with π, as glowacks began to note. Euler’s identity is really that exp(i)^(theta) = rotation by theta radians. Sure, the number of radians in -1 (half a revolution, 180 degrees) is π, but just as well, the number of radians in a complete revolution (360 degrees) is τ; exp(i)^τ = exp(iτ) = a complete revolution = (in terms of its rotational effect) 1. And this aspect of Euler’s identity comes up much more often in mathematics than that exp(iπ) = half a complete revolution = -1; the natural logarithm of a complex number is defined up to a multiple of τi, etc.
And just as well, the number of radians in a quarter revolution is τ/4, the number of radians in an eighth revolution is τ/8, etc. So exp(iτ)^(1/4) = a quarter revolution = i, exp(iτ)^(1/8) = an eighth of a full revolution = sqrt(i), etc. Surely, the observation “exp(iτ)^x = x many revolutions” is much more natural than “exp(iπ)^x = x many half revolutions”.
As for the area of a circle, for simple geometric reasons, the area of a circle is the area of a triangle with base equal to its radius and height equal to its circumference. I.e., 1/2 * C * r = 1/2 * τr * r. It happens to be the case that the 1/2 here cancels out with the τ into π, sure, but to really understand where the formula comes from is to appreciate where the 1/2 comes from (it’s the same 1/2 that appears in the formula for the area of a triangle, nothing more).