Understanding Pi: its Role in Symmetry and How it Was Found

[HallsofIvy]

Sorry to bring up something old, but:

For a circle of radius r, the circumference is equal to 4\int^r_0 \sqrt{ \frac{r^2}{r^2-x^2} } dx.

Not old enough! Werg22 asked about Euclid and that definition is much too recent for Euclid. Euclid did, in fact, define \pi as the ratio between the circumference of a circle and its radius.

 

[HallsofIvy]

What are you implying? The independence to analysis of geometry? Like I said, one needs not to introduce the definitions of analysis once results in geometry are derived, but I firmly believe that geometry needs to take ground on analytical definitions even though it was not always the case. I cannot find a meaning for circumference without referring to analysis and integration, if you can, by all means do tell.

Circumference of a circle certainly can be defined without integration- it just cannot be calculated exactly. It is true that Euclid "shorted" necessary concepts of continuity but Hilbert showed how to do that without analysis.

 

[Gib Z]

Not old enough! Werg22 asked about Euclid and that definition is much too recent for Euclid. Euclid did, in fact, define \pi as the ratio between the circumference of a circle and its radius.

Ahh No I was referring to the 2nd part of Werg22's post:

So how exactly is circumference defined in Euclid's terms? It is certainly not defined in terms of integration.

I thought perhaps the definition I offered would count as one in terms of integration :)

 

[Werg22]

I'm afraid I don't understand... I am advocating for the definition in terms of integration, not the opposite.

 

[Gib Z]

Ahhh Sorry! I read that wrongly >.< It could be read in two ways lol : 1. So how exactly is circumference defined in Euclid's terms? It (The Circumference) is certainly not defined in terms of integration..

Thats the one i thought, but you obviously meant

So how exactly is circumference defined in Euclid's terms? It (Euclid's definition) is certainly not defined in terms of integration.

 

[alfiejohn]

Pi is wrong

Has anyone seen the following?

http://www.math.utah.edu/~palais/pi.html

What I thought was clever about it was some of the examples:

- sin( x + thri ) = sin(x)
- e^(i.thri) = 1
- h bar = h/thri

where thri = pi with *three* legs, hence three

 

[Gib Z]

What is Pi with 3 legs?...All those equations would be valid if thri = 2\pi but somehow i don't think that's what you mean >.<

EDIT: I just read the link, and it turns out that IS what you mean ...His argument is true in the sense that the factor of 2 pops up repeatedly and often in many many formulas, but the reason pi is the way it is, is due to the fact that it originally came about in desire to find the area of a circle, and the pi without the factor of 2, is more useful in this sense, and more natural. But yes, I do agree factors of 2 accompany pi quite often, but in the end it makes no difference.

 

[alfiejohn]

It makes everything more elegant.

Kind of like why I use the decimal system. Roman numerals are just to clumsy ;)

 

[HallsofIvy]

What is Pi with 3 legs?...All those equations would be valid if thri = 2\pi but somehow i don't think that's what you mean >.<

EDIT: I just read the link, and it turns out that IS what you mean ...His argument is true in the sense that the factor of 2 pops up repeatedly and often in many many formulas, but the reason pi is the way it is, is due to the fact that it originally came about in desire to find the area of a circle, and the pi without the factor of 2, is more useful in this sense, and more natural. But yes, I do agree factors of 2 accompany pi quite often, but in the end it makes no difference.

I disagree with "it originally came about in desire to find the area of a circle". I think there is clear historical evidence that \pi was first used to find the circumference of a circle. Yes, one can easily write that "c= 2\pi r" using 2\pi, but if you are talking about a pillar or tree trunk, it is far easier to measure the diameter rather than the radius. That's why 'c= \pi d" is much more natural.

 

[ObsessiveMathsFreak]

I'm afraid I don't understand... I am advocating for the definition in terms of integration, not the opposite.

Why on Earth would anyone wish to do such a thing? What possible purpose or reason is there in denying at the outset what pi actually is?

It's simply not correct to come up with a formula or series that gives pi and take that as the definition, and later simply point out that the number also happens to be the ratio of the circumference to the diameter. That would be akin to coming up with a formula or series for Plank's constant or the magnetic permeability of a vacuum and declaring it as the definition for these constants.

Pi is the ratio of the circumference of a circle to its diameter. That's what it is. Millennia from now when no one is doing integration, or series or working in base ten, or using fractions, or just about anything we do now, pi will still be there as the ratio of circumferences to diameters. No number system, no analysis, no axioms, no definitions, nothing.

That's what pi is. How we find, measure or approximate it is entirely up to us, but the definition is quite out of our hands. You may be uncomfortable with this, but the universe does care about any philosophical objections you might have. It just is. Like pi.

 

[Integral]

I disagree with "it originally came about in desire to find the area of a circle". I think there is clear historical evidence that \pi was first used to find the circumference of a circle. Yes, one can easily write that "c= 2\pi r" using [it ex]2\pi[/itex], but if you are talking about a pillar or tree trunk, it is far easier to measure the diameter rather than the radius. That's why 'c= \pi d" is much more natural.

Timber cruisers in the woods of the Pacific Northwest use a tape measure marked in Pi units to measure the diameter of a tree. Just wrap the tape around the tree and read the diameter. At least they were used when there was a active logging industry.

 

[Werg22]

Why on Earth would anyone wish to do such a thing? What possible purpose or reason is there in denying at the outset what pi actually is?

It's simply not correct to come up with a formula or series that gives pi and take that as the definition, and later simply point out that the number also happens to be the ratio of the circumference to the diameter. That would be akin to coming up with a formula or series for Plank's constant or the magnetic permeability of a vacuum and declaring it as the definition for these constants.

Pi is the ratio of the circumference of a circle to its diameter. That's what it is. Millennia from now when no one is doing integration, or series or working in base ten, or using fractions, or just about anything we do now, pi will still be there as the ratio of circumferences to diameters. No number system, no analysis, no axioms, no definitions, nothing.

That's what pi is. How we find, measure or approximate it is entirely up to us, but the definition is quite out of our hands. You may be uncomfortable with this, but the universe does care about any philosophical objections you might have. It just is. Like pi.

The circle is not a physical entity, it's a mathematical one. The circle does not exist in nature for the simple reason that it is not applicable.

 

[Integral]

The circle is not a physical entity, it's a mathematical one. The circle does not exist in nature for the simple reason that it is not applicable.

nonsense, mankind has been making wheels for thousands of years. We have been making "perfect circles" , that is circles within our ability to measure, for thousands of years.

Whether or not a perfect circle exists is irrelevant.

 

[Werg22]

A ten year old might qualify a wheel a circle, I do not.

 

[Integral]

That is your problem not mine.

Thread done.