Pi day! (3/14/09)

[fluffy]

Well yeah, that was the point to drew's cartoon, and it's not like anyone seriously needs the accuracy of pi for precise measurements or whatever. But it's a fascinating number and it's easy to accidentally memorize numbers if you're exposed to them constantly, like if they're printed on a banner in a classroom where you're normally bored.

Other numbers I have memorized without intending to: my credit card number (including CVV2 code), my badge unlock code for the fingerprint scanner at work, and dozens of different license plate numbers from cars over the years. I'd certainly prefer my neurons to be trained to more useful things, but oh well. On the plus side I never have trouble remembering my ATM PINs, at least.

[Ross]

Just as a FYI, according to the wiki article:

a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.

So, in common use 3.1415927 is more than enough. Unless you're calculating the circumference of the known universe.

In which case you're also assuming the universe is finite, euclidian, and circular - brash assumptions.

[Billy's Little Trip]

Of course, when I started the thread, this is the comic I had in mind, and I am sad nobody linked to it:

Actually, it should look like this.

[Spud]

SpudNut has 17 digits past the decimal point memorized.

[JonPorobil]

If that's true, then how did did scientists figure it out beyond 39 digits?

[Märk]

Well yeah, that was the point to drew's cartoon, and it's not like anyone seriously needs the accuracy of pi for precise measurements or whatever. But it's a fascinating number and it's easy to accidentally memorize numbers if you're exposed to them constantly, like if they're printed on a banner in a classroom where you're normally bored.

Other numbers I have memorized without intending to: my credit card number (including CVV2 code), my badge unlock code for the fingerprint scanner at work, and dozens of different license plate numbers from cars over the years. I'd certainly prefer my neurons to be trained to more useful things, but oh well. On the plus side I never have trouble remembering my ATM PINs, at least.

It is an amazing number, for sure. The square root of 2 is another one. (and part of the decimal sequence of that appears in pi, oddly enough)
I memorized the alphabet backwards. It's easy, when you know how to do it, the trick is that it actually rhymes better backwards:
Z Y X, W V (Zed why ecks, double you vee)
U T S, R Q P (You tee ess, are queue pee)
O N M, L K J (Oh enn emm, ell kay jay)
I H G F (Eye aitch gee eff)
E D C B A (EeeDeeCeeBee Aye!)

[Billy's Little Trip]

Would you guys stop talking about stupid insignificant numbers and tell me my picture is funny?

.....apple! GET IT!

I slave over a hot computer all day and all I ask is a little appreciation. Is that too much too ask? Maybe a "thank you dear" or "yes honey poo, that was wonderful" Would it kill you to pull yourself away from your "preciouses" long enough to smell my roses!?!

....You know what, never mind. No, just never mind. If you need me, I'll be at my mothers.

















.....annnnnd cut! That's a wrap, put it in the can.

[Spud]

Would you guys stop talking about stupid insignificant numbers and tell me my picture is funny?

Actually, I didn't get it. I didn't think you had a mac, so I couldn't figure out the apple thing. It finally dawned on me that you were talking about apple pie. Perhaps you should have gone with mincemeat or lemon meringue or something in order to make the joke more obvious and therefore funny.

[fluffy]

If that's true, then how did did scientists figure it out beyond 39 digits?

I can't tell if you're being serious or just making a joke about it being the ratio of the circumference to the diameter.

If you're being serious, there are many ways to derive pi via various convergent series. The easiest one to explain is probably just finding the area of a circle of radius 1, by cutting the curve between sqrt(1-x^2) and -sqrt(1-x^2) into smaller and smaller rectangles.

[fluffy]

t is an amazing number, for sure. The square root of 2 is another one. (and part of the decimal sequence of that appears in pi, oddly enough)

Not particularly odd. Considering the high entropy in pi*10^x mod 10, it's inevitable that you'd find pretty much any sequence of digits somewhere within Pi. Try it yourself!

[Billy's Little Trip]

Would you guys stop talking about stupid insignificant numbers and tell me my picture is funny?

Actually, I didn't get it. I didn't think you had a mac, so I couldn't figure out the apple thing. It finally dawned on me that you were talking about apple pie. Perhaps you should have gone with mincemeat or lemon meringue or something in order to make the joke more obvious and therefore funny.

Damn! Mincemeat crossed my mind, but I didn't know how to spell it! It was originally cat, but I didn't want people thinking I was stupid.

[Märk]

Here's something I just tried, and maybe my math is wrong, but something is odd.

Okay, say the area of a circle is exactly 625 square units.

Stretch the circle into a perfect square shape, and the square's area should be identical, 625 units square, or 25 X 25 along the sides, meaning the circle's circumference *should* be exactly 100, but it's not; (625/pi)/2 = 99.471839432434584855552352107821... which is very close, but it's not 100. In fact, using my computer's calculator, which calculates pi to 31 decimal places after the 1, it should be a lot more accurate. What is going on here?

[Lord of Oats]

Well, like you said, you're stretching it. Or destretching, in your case.

Set up a 2-D model with a piece of string and take measurments. If your math is correct here, then a different shape with equal circumference will yield an unequal area.

[Märk]

I think what the problem is, is that pi is actually exactly 3.125. Damned math people have been lying to us all these years!

(625/3.125)/2 = 100

[edit] Okay, this is weird and inconsistent- suppose that the circumference of the circle/perimeter of the square is 100.53096492 units. (sides of 25.13274123 units) This gives an area of 631.6546817341419129 for the square, and.... (631.6546817341419129/3.1415926535897932384626433832795)/2=100.5309649251266163694568455671.. accurate to 8 decimal places. Something is really fucking weird here.

[Lord of Oats]

Irrational numbers just make me think that there must be something wrong with our mathematical system. For instance, I'm pretty sure we need to adjust our numbers so that pi = 3. That would make things a lot easier. Another possibility is to make it so 1 is the square root of 2.

[fluffy]

Here's something I just tried, and maybe my math is wrong, but something is odd.

Okay, say the area of a circle is exactly 625 square units.

Stretch the circle into a perfect square shape, and the square's area should be identical, 625 units square, or 25 X 25 along the sides, meaning the circle's circumference *should* be exactly 100, but it's not; (625/pi)/2 = 99.471839432434584855552352107821... which is very close, but it's not 100. In fact, using my computer's calculator, which calculates pi to 31 decimal places after the 1, it should be a lot more accurate. What is going on here?

http://en.wikipedia.org/wiki/Squaring_the_circle

How exactly are you "stretching" it?

[Märk]

Here's something I just tried, and maybe my math is wrong, but something is odd.

Okay, say the area of a circle is exactly 625 square units.

Stretch the circle into a perfect square shape, and the square's area should be identical, 625 units square, or 25 X 25 along the sides, meaning the circle's circumference *should* be exactly 100, but it's not; (625/pi)/2 = 99.471839432434584855552352107821... which is very close, but it's not 100. In fact, using my computer's calculator, which calculates pi to 31 decimal places after the 1, it should be a lot more accurate. What is going on here?

http://en.wikipedia.org/wiki/Squaring_the_circle

How exactly are you "stretching" it?

Certainly not with a compass and ruler! An imaginary line of a given length, formed into a circle, then the same line formed into a square. I did say my math might be wrong (I was never good at geometry) but it seems like the area should be the same for both shapes... and it almost is, but using pi to calculate the radius from the (assumed) area of the circle comes up short. However, when using 631.6546817341419129 as the starting area, it works out exactly to 8 decimal places. Is this a paradox, or am I making a schoolboy error in the relationship of area to squares and circles? Trust me, I googled till my eyes were sore, and can't find any answers.

[fluffy]

I am certain you are making an error but I'm still not at all sure what the hell it is you're doing.

How are you determining the length of the square's side? Also, it's quite likely that there is a rounding error in the calculator - just because it can display N digits of useless precision doesn't mean it actually calculates with that much precision (especially if it internally uses base-2 instead of base-10).

Many calculators (such as TI's) actually use base-10 (BCD) math for everything to reduce base-conversion precision errors, which are especially nasty for floating point, but ones which do that also don't pretend to have more precision than they actually do.

Anyway, 8 digits of floating-point precision is pretty good and really about all you can expect, if it's doing actual binary floating-point. A 32-bit binary floating point number usually has 8 bits of exponent and 24 bits of mantissa, meaning it's only precise to about 1/16777216 of the magnitude of the number (for example, the number 123.4567 will be precise to within about .0000073585). It depends on the internal representation and so on though.

[Spud]

Why would the area be the same for both shapes? This may be intuitive, but it is also wrong.
The ratio of perimeter to area is hardly a constant. Take a look at these two figures:



Both have a perimeter of 14, but one has twice the area of the other.

[fluffy]

Wait, is that what Mark was doing? Yeah, that's completely wrong. I'm surprised the number was as close as it was.

[Spud]

BTW, the same goes for surface area to volume. It is not a constant ratio. Otherwise, toothpaste tubes wouldn't work.

Here's a weird one for you, though: what is the ratio between the length of the side of a cube and its area?

Suppose we start with a unit cube, 1 unit on a side. The area is one. so the ratio is 1:1
If the cube is 2 units on a side, the area is 4, so the ratio is 1:2
if the cube is 3 units on a side, the area is 9, so the ratio is 1:3
and so on...

That's for a freaking square! The shape is constant! do do do do... do do do do...

[Märk]

Why would the area be the same for both shapes? This may be intuitive, but it is also wrong.
The ratio of perimeter to area is hardly a constant. Take a look at these two figures:



Both have a perimeter of 14, but one has twice the area of the other.

I'm only using circles and squares though, both equilateral shapes. To put it another way:

(c=circumference, r=radius, a=area, p=perimeter)
circle: c 100, r 15.915494309189533576888376337251, a 628.3185307179586476925286766559
square: p 100 (25 X 25), a 625

BUT

circle: c 100.53096492, r 16.00000000081592633649327484316, a 631.6546817341419129
square: p 100.53096492 (25.13274123 X 25.13274123), a 631.6546817341419129... identical, to 16 decimal places!

How can the area of a circle as compared to the area of a square, using exactly the same circumference/perimeter not always be consistent? I'm not saying they should be the same as each other, but the relationship should always be consistent, but I've just demonstrated that it is not.