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

[sutupidmath]

well, i can't say much about this, however in some of the formulas in analysis that i have seen, the formulas which are taken to be definition of pi, i think that the concept of pi is already involved in them. I say this because in most of the proofs, it starts from trig functions, and later on involve pi, for example François Viète's formula etc.
As i said i know very, very little about this.

 

[Gib Z]

For your research, it is known as the Symmetric formula, found by Jonathan Sondow.

 

[cristo]

Yes I am very well conscious that it was known way before. However, I believe you reckon that modern analysis redefined all geometry known before its elaboration - and modern analysis wants pi to be derived out of the concept of integration.

This sentence that you're putting into my mouth doesn't make sense!

It's as if all mathematics prior to analysis had never existed. Before analysis, the proportionality of circumference to radius was known because of our physical interpretation of distances and areas, but with analysis, this proportionality is a result independent of physical reality.

You appear to be digressing from the subject. The original question was what is pi and how was it discovered. I gave the standard, natural definition to do with geometry which is clearly the origin of the constant. I don't see why you're arguing this anymore-- ok, so there is a definition of pi grounded in analysis, but this was definitely not the origin!

 

[Feynman diagram]

Perhaps the thing I find most amazing is the variety of situations in which we encounter Pi in theoretical physics. Everything from the Scrodinger equation to n-body differential phase space formulae and beyond, to the extent that any relation to geometric interpretations is lost. We could debate all day about where the fundamental origin of Pi is, but at a basic level I would be happy with the circle circumference relation, and wait until I had learned real analysis to make up my own mind whether or not I thought the definition in terms of convergent series was better, or indeed any other definition anyone cares to provide.

 

[Gib Z]

www.wikipedia.org/wiki/Pi has quite a few definitions.

 

[Werg22]

Any word I wish to use to describe that statement would be asterisked out - there is a whole world of mathematics that has nothing to do with, and no dependence on analysis.

Sorry, I should have said "plane geometry" instead of "mathematics".

 

[Werg22]

You appear to be digressing from the subject. The original question was what is pi and how was it discovered. I gave the standard, natural definition to do with geometry which is clearly the origin of the constant. I don't see why you're arguing this anymore-- ok, so there is a definition of pi grounded in analysis, but this was definitely not the origin!

Certainly it is not the origin as a constant known to man, but analysis allows pi to be defined independently of the its physical interpretation. Arc length and area should now be regarded as definitions that are purely analytical grounded on the concept of integration. From that point of view, pi derives from the analytical foundation of those two definitions; pi is introduced for the "first" time when one looks at the analytical definition of area and circumference - this is what I mean by "origin".

 

[ZioX]

Was Euclid able to prove that the ratio of the circumference and the diameter is always pi from his axioms?

How was he able to define circles while lacking coordinate geometry?

 

[Werg22]

As stated, Euclid probably saw the circle as an abstract idea. His definition of length and segment was also an abstraction. However Euclid could not prove mathematically that the ratio between circumference and diameter is constant - nor did he even think about the question since the concept of circumference was not known to him outside of physical representation.

 

[matt grime]

Sorry, I should have said "plane geometry" instead of "mathematics".

My comment still holds - plenty of geometry is done and is independent of any and all analysis.

 

[HallsofIvy]

Werg22, I certainly agree with much of what you are saying- pi can be defined completely analytically (as half the period of sine, for example, which can itself be defined "non-geometrically"). However, your original statement was "Yes pi is the ratio of the circumference to the diameter, however the "origin" of pi, how it is discovered and defined, is not there."

Perhaps, since you put "origin" in quotes you meant its basic definition in analysis, from which, using analytic geometry, derive the fact that the pi is the ratio of the circumference of a circle to its radius. However, since the original question had to do with the historical origin of pi, that use of the word is at best confusing.

 

[Werg22]

My comment still holds - plenty of geometry is done and is independent of any and all analysis.

Yes, but all of the foundations of plane geometry can be said to pertain to analysis; the definition of distance, area, sine, cosine, etc.. This said, once the results are known, we need not to mention the concept of integration every time we have to calculate the area of a triangle or the circumference of a circle.

HallsofIvy, I reckon that the use of the word origin could be confusing. I will have to specify; origin here means the origin at which modern mathematics bases the number pi on. One needs not to look at the history of pi; as said, we should not concern ourselves with the geometry that was used before analysis - analysis should be seen as the basis of all euclidean geometry.

 

[matt grime]

Yes, but all of the foundations of plane geometry can be said to pertain to analysis; the definition of distance, area, sine, cosine,

nope. can't buy any of that at all. it almost seems that you're reasoning in the diametrically opposite direction to which you're claiming.

 

[Werg22]

Matt, we are perhaps irreconcilable. The point I am trying to pass is that analysis does not need the geometry used prior its elaboration to give us all the results of classical geometry. However, my view is that geometry, prior to analysis, was not satisfactory in the firmness of its definitions; analysis makes circumference a purely mathematical idea while without it, circumference can only be defined physically.

 

[DeadWolfe]

analysis makes circumference a purely mathematical idea while without it, circumference can only be defined physically.

Nonsense. Euclid's geometry is as mathematical as anything that predates set theory.

 

[Werg22]

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

 

[matt grime]

However, my view is that geometry, prior to analysis, was not satisfactory in the firmness of its definitions; analysis makes circumference a purely mathematical idea while without it, circumference can only be defined physically.

Have you actually taken any courses on geometry?

 

[Werg22]

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.

 

[rcgldr]

Regarding the origin of "pi", it was measured by objects that approximated circles. A tape could be wrapped around a circular object and compared to the distance across the object. A wheel could be revolved one revolution (this is actually done to get effective circumference of automobile tires, drive over a thin wet strip and measure the distance between the strips created by a moving tire). It's also common to rate tire circumferences as revolutions per mile tire rack specs for some tires .htm

It wouldn't take a lot of measurements to realize that the ratio between circumference and diameter was a constant for wheels and/or circular objects of any size.

So the measurement of pi was originally a physical exercise. The abstract concept of a perfect circle and diameter are mathematical in nature, and there are various mathematical methods for calculating pi.

I still prefer to define pi as the ratio of circumference to diamenter. After all, this is the basis for angular units called radians. Why all the effort to turn a simple concept into something unnecessarily complicated?

On an analog computer, generating sine waves with a period of 2*pi is no problem, simply set the 2nd derivative of Y to -Y, and set initial values for the first derivative of Y and Y and let it rip (analog computers integrate over time).

 

[Gib Z]

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

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.

 

[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.