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

[Mr Virtual]

Hi all
In almost all the formulae related with symmetry, we use pi. For example, in case of circle, we use pi, and in case of sphere we use 4*pi.
e.g. E (electric field)= (q * qo) * 1/(4*pi*eps* r^2)
A (circle)= pi*r^2
A (sphere)=4*pi*r^2

My question is: How does pi signify symmetry? Why is the value of pi what it is, and what does this value (22/7) represent? How was this value originally found?

regards
Mr V

 

[morphism]

pi is not 22/7 - this is merely an approximation.

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

 

[Werg22]

Pi could be said to be the convergence of series that we obtain out of some integrals.

 

[cristo]

pi is the ratio of the circumference of a circle to its diameter.

 

[Werg22]

pi is the ratio of the circumference of a circle to its diameter.

I purposefully avoided that definition to stay away from any physical interpretation of pi. Modern analysis would define the concept of integration before circumference, making the definition in question a result rather than a definition in itself.

 

[d_leet]

Pi could be said to be the convergence of series that we obtain out of some integrals.

What exactly does this mean?

 

[cristo]

I purposefully avoided that definition to stay away from any physical interpretation of pi. Modern analysis would define the concept of integration before circumference, making the definition in question a result rather than a definition in itself.

But pi is a fundamental constant of nature (along with G, c, etc), and thus is a physical quantity. The value of this constant is determined by the ratio of the circumference of a circle to its diameter, for any circle.

Besides, how were you taught about pi? I definitely didn't know how to integrate before the concept of pi was introduced. It is introduced in geometry classes in school.

 

[Mr Virtual]

Thanks for your replies

Mr V

 

[Werg22]

What exactly does this mean?

It simply means that pi's definition is derived out of the concept of integration; that is the function f(x) = \sqrt{r^{2} - x^{2}}, whether we consider its circumference or area, both defined in terms of integral nonetheless, pi is defined within those limits. Out of the integrals that define pi, we can derive series that converge towards pi as defined.

 

[Werg22]

But pi is a fundamental constant of nature (along with G, c, etc), and thus is a physical quantity. The value of this constant is determined by the ratio of the circumference of a circle to its diameter, for any circle.

Besides, how were you taught about pi? I definitely didn't know how to integrate before the concept of pi was introduced. It is introduced in geometry classes in school.

But the introduction of pi in school is the presentation of a result: that the circumference of any circle is directly proportional to its radius. 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.

 

[cristo]

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.

Erm... the fact that the ratio of the circumference to the diameter of a circle is equal to some constant for all circles was known long, long before calculus was even invented. The exact value of the constant may have been calculated by taking limits of integrals, but the OP asked for the origin of this constant. It most certainly originates from the definition I gave.

 

[Werg22]

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. 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.

 

[damoclark]

Pi

From a geometric point of view, Pi is what is it due to how unit area is defined.
It's just so happens that on planet earth, humans decided to define a unit of area as a little square box with sides of unit length.

In another corner of the galaxy, some other intelligent life form might have decided to define unit area as a little circle with unit diameter, in which case their circles would have an area of 4*r^2, rather than Pi*r^2

 

[robert Ihnot]

Good Heavens! If anybody took Plane Geometry like in the old days, you would know that Euclid defined that. The critical fact is that it relates all circles to their diameters, and thus is a justified constant.

Werg22: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.

Actually the Greeks recognized that the "perfect circle," was an abstraction, which they termed an "Ideal." And there has been a long belief in the existence of Ideals apart from physical forms. Decartes wrote at length on this, and even Godel was a believer in Platonism. Godel I quote:

Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.
It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e., in the latter case the actually occurring sense perceptions. http://www.friesian.com/goedel/chap-2.htm

 

[Werg22]

Good Heavens! If anybody took Plane Geometry like in the old days, you would know that Euclid defined that. The critical fact is that it relates all circles to their diameters, and thus is a justified constant.

Yes Euclid did define the ratio but he did so because, at the peril of repeating myself, because of the physical interpretation of what circumference and diameter are. As I said, it's as if anything prior to analysis never existed, every definition that we now have has to be defined by analysis and nothing else.

 

[robert Ihnot]

You got in there before I could finish: Godel I quote:

Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.
It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e., in the latter case the actually occurring sense perceptions. http://www.friesian.com/goedel/chap-2.htm

WErg22: As I said, it's as if anything prior to analysis never existed, every definition that we now have has to be defined by analysis and nothing else.

I would love to know who thinks that? Can you name a single outstanding mathematican who says that? I don't think it was Godel.

 

[rcgldr]

pi is the ratio of the circumference of a circle to its diameter.

I purposefully avoided that definition to stay away from any physical interpretation of pi.

... but that definition is mathematical, not physical. Definition of circle and diamter are mathematical. There are some physical objects which approximate a circle, but they're not perfect.

Defining a series to represent pi doesn't mean that pi is defined by the series. I thought the goal of most sciences is to use the simplest definitions as long as they are accurate.

 

[robert Ihnot]

Jeff Reid: Defining a series to represent pi doesn't mean that pi is defined by the series. I thought the goal of most sciences is to use the simplest definitions as long as they are accurate.

Sounds good to me! However, I suppose if you are working in analysis, you might get your own set of ideas, but if you were in Algebra, maybe you have different ideas.

 

[Werg22]

robert Ihnot: In the ideal circle pi is the ratio between circumference and diameter. However, what is circumference? The Greeks only had physical means to define it. They may have believed that a the circle was ultimately an abstract idea, but did not have any tool to set that intuition into stone. All that I am saying is that I am convinced that the starting point shouldn't be the ratio itself but rather what circumference (or area) is and how is it mathematically represented - pi arises from that starting point. However I realize this is a egg-chicken question and I myself admit there is some room for debate.

 

[robert Ihnot]

I once heard in Physics that if there are two competing theories to use with both giving satisfactory results, then prefer the simplest.

Then again some will argue that anything out of Physics could not be valid here, and perhaps the more complicated, advanced, difficult theory should be chosen. After all, otherwise maybe too many people would pass math.

 

[Werg22]

I am still learning allot and am trying to get a hold of the philosophy of mathematics. The ideas that I have expressed have been the conclusions I have reached after thinking about those questions for a prolonged period of time. Maybe it's just like any other "philosophy"; it cannot be said to be "better" than another.

 

[robert Ihnot]

Werg22: All that I am saying is that I am convinced that the starting point shouldn't be the ratio itself but rather what circumference (or area) is and how is it mathematically represented - pi arises from that starting point. However I realize this is a egg-chicken question and I myself admit there is some room for debate.

Well, no matter how you set it up, you are going to have to use axioms and develope a system, so that only a limited degree of truth can be revealed by such a system, as Godel has proven. Some people do not feel that foundations has been a successful subject of investigation. One writer wrote a whole book on meaning of "1." Same author later decided that the foundations of math were tottering.

I once heard a logic professor talk about a picture, he said that some believed it was O.K. to draw a picture, but not to refer to it in a proof. I don't know what his actual opinion of that sort of situation was, but knowing his nature he hated philosophy and questions like, "What is the difference between the morning and evening star." So he probably disaproved of making things harder and more involved than they really were.

Anyway, you have not named a Mathematician or mentioned a school of thought that upholds your opinion.

 

[robert Ihnot]

Werg22: I am still learning allot and am trying to get a hold of the philosophy of mathematics. The ideas that I have expressed have been the conclusions I have reached after thinking about those questions for a prolonged period of time. Maybe it's just like any other "philosophy"; it cannot be said to be "better" than another.

Well, its a highly commendable try anyway. I guess you need advice from some prof who could help you develope such ideas.

 

[Werg22]

Anyway, you have not named a Mathematician or mentioned a school of thought that upholds your opinion.

I do not know if anyone shares my opinion, nor have I tried to find out. But I am not interested in knowing that... as long I am satisfied with my understanding of things.

 

[eigenglue]

pi is equal to the natural logarithm of -1 divided by the square root of -1.

 

[matt grime]

As I said, it's as if anything prior to analysis never existed, every definition that we now have has to be defined by analysis and nothing else.

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.

 

[Gib Z]

pi is equal to the natural logarithm of -1 divided by the square root of -1.

That is somewhat true, although the natural logarithm of -1 is actually i\pi +2ki\pi, k \in \mathbb{Z} and there are 2 square roots of -1, i and -i. Avoiding pedantic trivialities, that is not really a helping way of explaining to a newbie what pi represents.

I personally think that the definition Cristo gave is the best, the series Werg22 talked about arose from the first simple definition of pi given by Cristo. Many definitions can be proven to be equivalent, with some being more rigorous and room for error than others, but seriously anything more complex than "the lowest positive real x for which cos x equals -1" is getting out of hand.

 

[mahoutekiyo]

can't both Werg and Robert be correct in this case? I mean... isn't the integral formula for pi just a proof of its abstract physical measure? ='\

 

[Gib Z]

The argument wasn't over the correctness of their definitions, but more so the simplicity and therefore naturalness of each definition. I could provide you with an exceedingly difficult definition of pi, such as the value that the following converges to:

\frac {\displaystyle \prod_{n=1}^{\infty} \left (1 + \frac{1}{4n^2-1} \right )}{\displaystyle\sum_{n=1}^{\infty} \frac {1}{4n^2-1}} = \frac {\displaystyle\left (1 + \frac{1}{3} \right ) \left (1 + \frac{1}{15} \right ) \left (1 + \frac{1}{35} \right ) \cdots} {\displaystyle \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \cdots}

As you can see, that was extremely impractical and didn't help the OP in his actual understanding of "What is pi?"

 

[mahoutekiyo]

well, that's what I meant. Aren't they both rather natural but just in different contexts. I dunno. I think it would be important to see both the physical sense of pi and the abstract sense of pi as very important to learning the answer to the question, "what is pi"... :: shrug ::

and now I will go and research that grandiose summation you posted ^ ^

 

[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.