Irrational Pi Defrocked

[C1ay]

Yes, I have, Clay. And where do you get the notion that the finite pi iis 1000's of years old? It was arrived at only this last year. Let's cut to the chase here ....prove for us that the irratrional pi value of 3.14159....is more accurate than that of, say,3.16409...ad infinitum.

The burden of proof is yours. Read the FAQ, # If you make strange claims, please provide proof. Your claim is not just strange though, it defies all modern math texts.

[MortenS]

It is a bit difficult to prove it to you, robust, as I do not know what kind of math you understand nor accept.

 

One simple way to get better and better approximations of pi, is to put regular polygons inside and outside a circle, and then increase the number of sides. As the number of sides in a polygon goes towards infinity, the polygon gets closer and closer to become a circle. If you calculate the ratio of the circumference of the polygons and their diameters, you will get better and better boundaries where pi will lie between.

 

In a polygon with n sides, and radius r, the circumference is given by the formulae:

 

n*2*r*tan(360/n)

 

let r=1 --> diameter = 2

 

For n= 4 (a square):

circumference/diameter= 4

For n = 6

circumference/diameter = 3,464101615138

 

for n=12

circumference/diameter = 3,215390309173

 

for n = 20

circumference/diameter = 3,167688806491

 

for n = 30

circumference/diameter = 3,153127057970

 

for n = 100

circumference/diameter = 3,142626604335

 

for n = 1000

circumference/diameter = 3,141602989056

 

for n = 10000

circumference/diameter = 3,141592756944

 

for n = 100000

circumference/diameter = 3,141592654644

 

What you see, when you increase the number of sides in the polygon, is that the ratio between the circumference of the polygon, and its diameter will approach pi= 3.141592.....

 

In fact, you can define a circle as a polygon with an infinite number of sides.

 

Therefore, if you agree that pi is defined as the ratio between the circumference of a circle and its diameter, regardless of what actual value this ratio actually has, you must agree that it has to lie between the ratio of circumference/diameter of the inner polygon you can inscribe in a circle and of the outer polygon of a circle.

 

You must also agree that as the number of sides in the polygons increase, the closer the polygons circumference will come to the circles circumference.

 

It follows from this, that if you measure the circumference of polygons with increasing number of sides, and divide by their diameter, you will get closer and closer to the true value of pi.

 

Please disprove that this is not the case....

[Robust]

You put forth a valid arguement with your polygon example, Moton S. The problemwith it I see is that you are equating the polygon lines with that of the radius. It doesn't (cannot) work that way for the simple reason that there is no such thing as the straight line. That's why the "straight" line of the radius is equated with the curved line of the radian; thus: radius/radian gives the distance between each angular degree on the curved circumference. You will never get the correct area of the circular plane factoring line by line......IMHO.

[MortenS]

Let me find a drawing that shows you what I mean, then it may be easier to see that the r in the polygons are actually the same r as in the circle (and even the diameter, at least in some polygons). In this case, the polygon is inscribed in the circle, so it is logical that the circumference of the polygon is smaller than that of the circle.

 

It is also logical, that as you increase the number of sides in the regular polygon inscribed, you will get closer and closer to the circumference of the circle.

 

 

Source: http://acm.uva.es/problemset/v104/10432.html

[Buffy]

You put forth a valid arguement with your polygon example, Moton S. The problemwith it I see is that you are equating the polygon lines with that of the radius. It doesn't (cannot) work that way for the simple reason that there is no such thing as the straight line. .... You will never get the correct area of the circular plane factoring line by line......IMHO.

I have not followed this thread much but the highlighted quote here really got me interested in reading all 13 pages of this thread. Facinating bit of discussion not to unlike some of the other metaphysical discussions here. To recap, its seems there are four key issues that invalidate robust's arguments

1. robust is using a formula thats simply a tautological equation that lets you put in any number you want for pi and the same area/circumference comes out for given values of radii. In reading all the posts here, there was no explanation why this equation is relevant to anything and it is not derivable from any of the relevant equations for determining dimensions of a circle. This equation could be used to argue that the value of pi could be 1 or 12 or 453 or google for that matter.
   
2. there are numerous indications, pointed out by rince at one point, that robust does not understand the difference between an approximation and an actual value. indeed I get the sense that the whole point of robust's argument is that we can't prove that any particular approximation of pi is or isn't wrong, because it can't be measured exactly. Morten's most recent post of course is based on calculus, and by robusts response in the quote I've included to this one and the early post by clay about carrying out the various algebraic approximations to an infinite number of terms, he does not understand the notion that these methods of approximation basically provide a mechanism by which those "straight" lines get closer and closer to the curve of the circle, and provide more precision to more decimal places of the area/circumference. I get the feeling he's seen references to these methods of approximating pi to any precision in his references to Euler and others, but that he doesn't understand or accept them.
   
3. related to this, robust does not seem to accept that if an area is a rational number (like 16) that the radius might be irrational, or that pi could be irrational, or more importantly that might have a decimal representation that continues to an infinite number of decimal places. He occasionally uses the term "finite" which as was also pointed out in mathematical terms means that it is not infinte, although robust seems to use it to mean that a number has a decimal representation to a finite number of decimal places, which of course points back to the original source of this as being an engineering problem which of course would end up having a decimal solution to the fixed precision of the computing device used, and would to mathematicians mean its an approximation and any formulas based on these approximations would yield an approximate pi, not pi.
   
4. there are no references to any other sources he's given, although some have been found by other correspondents here, that would provide any evolution of the use of the equation that is clearly irrelevant, but is at the heart of his arguments, and despite numerous entreaties, he has not discussed the relevance of it or how it was derived.
   

I guess it will be interesting to see how much further this one goes, but I'm starting to get dizzy. Its obvious that many of our math experts here have chosen not to participate in this discussion, and that's probably a good indication of how much the experts find it pointless to say anything. Since robust points out he's not a "maths person" it makes you wonder why he's so sure of his conclusions.

 

So robust, can you tell us all why an area of mathematics that has not been debated in millenia was overturned "just in the past year" and none of the many mathematicians resident here has ever heard of it?

 

Cheers,

Buffy

[MortenS]

Robust, I am going to go back to the figure I just posted, and ask you to imagine a few things.

 

First let me know if you agree with the following statement:

 

1. The circumference of the inner polygon is less than the circumference of the circle, and it will allways be.

 

2. If an outer polygon, with the same number of sides was added, it would have a circumference larger than the circle, and it would allways be larger.

 

If you agree with this, and we let cpi be circumference of the inner polygon, and cpo be the circumference of the outer polygon and cc be the circumference of the circle, I guess you would agree with me on the following:

 

cpi < cc < cpo

 

if this relation is true, it follows that if we divide on 2*r (which is the diameter of the circle) we will get the following:

 

cpi/2r < cc/2r < cpo/2r

 

Whether you agree with me on the method of calculating the circumference of the polygons or not, I am sure that you agree with me so far.

 

If you do not agree with me on this, we have some more rounds to go before I proceed on how to calculate the circumference of the polygons(besides measuring it, which is easy to do when the number of sides is not that large...when the number of sides increase to a very large number, you need to find a method to calculate the lenght of the sides, since it becomes impractical to measure after a while).

 

Just let me know if you agree so far, and I will take it to the next level from here. I need to understand exactly where you mean I am wrong.

[Robust]

It is a bit difficult to prove it to you, robust, as I do not know what kind of math you understand nor accept.

 

One simple way to get better and better approximations of pi, is to put regular polygons inside and outside a circle, and then increase the number of sides. As the number of sides in a polygon goes towards infinity, the polygon gets closer and closer to become a circle. If you calculate the ratio of the circumference of the polygons and their diameters, you will get better and better boundaries where pi will lie between.

 

In a polygon with n sides, and radius r, the circumference is given by the formulae:

 

n*2*r*tan(360/n)

 

let r=1 --> diameter = 2

 

For n= 4 (a square):

circumference/diameter= 4

For n = 6

circumference/diameter = 3,464101615138

 

for n=12

circumference/diameter = 3,215390309173

 

for n = 20

circumference/diameter = 3,167688806491

 

for n = 30

circumference/diameter = 3,153127057970

 

for n = 100

circumference/diameter = 3,142626604335

 

for n = 1000

circumference/diameter = 3,141602989056

 

for n = 10000

circumference/diameter = 3,141592756944

 

for n = 100000

circumference/diameter = 3,141592654644

 

What you see, when you increase the number of sides in the polygon, is that the ratio between the circumference of the polygon, and its diameter will approach pi= 3.141592.....

 

In fact, you can define a circle as a polygon with an infinite number of sides.

 

Therefore, if you agree that pi is defined as the ratio between the circumference of a circle and its diameter, regardless of what actual value this ratio actually has, you must agree that it has to lie between the ratio of circumference/diameter of the inner polygon you can inscribe in a circle and of the outer polygon of a circle.

 

You must also agree that as the number of sides in the polygons increase, the closer the polygons circumference will come to the circles circumference.

 

It follows from this, that if you measure the circumference of polygons with increasing number of sides, and divide by their diameter, you will get closer and closer to the true value of pi.

 

Please disprove that this is not the case....

MortonS, I wouldn't attempt to prove or disprove your polgon exercise, particularly as extending beyond the circle. I do know in that regard, however, that a square describing the cardinal points of the circle is by the formula radius*sqrt 2 as giving the side of a square, the vertices of which define the 4 cardinal points of the circle, the area of which is 1/2 pi that of the circle. I don't really know how all this relates. I'm merely attempting to show that the irrational pi is not sacrosanct.

 

I have shown that the "irrational" pi of no greater purpose or accuracy than any of the other known pi vaues I am inclined to say in that regard, however, that any pi value incapable of describing the closed continuum of the circle by other than a whole number or ending decimal is to be disregared....particularly in considering the cosmologicasl navigation plans we ihave n the making.

[tom]

I have shown that the "irrational" pi of no greater purpose or accuracy than any of the other known pi vaues I am inclined to say in that regard, however, that any pi value incapable of describing the closed continuum of the circle by other than a whole number or ending decimal is to be disregared....particularly in considering the cosmologicasl navigation plans we ihave n the making.

 

No you haven't. That you call a demonstration is a statement wich is true for every number ( except 0) and that's not a demonstration.

 

 

That do you say about this formula, it's like yours

 

pi + 16 - a - pi +2*a = a + 16

[Rincewind]

... I'm merely attempting to show that the irrational pi is not sacrosanct.

 

I have shown that the "irrational" pi of no greater purpose or accuracy than any of the other known pi vaues...

I don't know how I can put this simpler. Pi is the ratio between the circumference and diameter of a circle and can only have one value. All the rest are poor approximations, especially your 3.16whatever.

 

Sacrosanct means sacred. There is nothing sacred about the correct value of pi -- it just is the correct value. This is not about religion.

 

Buffy has explained very clearly (I thought) why any value (even ridiculous ones) works in your circular formula -- it just cancels pi out from the whole equation, and, as such, shows absolutely nothing about pi.

 

There is, however, only one correct value of pi, and MortenS has shown you the best way to get far closer to it than the "other known pi values" manage.

 

Try it yourself, Robust -- "do the math."

[C1ay]

I have shown that the "irrational" pi of no greater purpose or accuracy than any of the other known pi vaues I am inclined to say in that regard, however, that any pi value incapable of describing the closed continuum of the circle by other than a whole number or ending decimal is to be disregared....particularly in considering the cosmologicasl navigation plans we ihave n the making.

No you haven't. It is things like cosmological navigation that require greater and greater approximations of pi, not obtuse values like your so called finite pi.

[Robust]

No you haven't. It is things like cosmological navigation that require greater and greater approximations of pi, not obtuse values like your so called finite pi.

 

Then tell us, Clay, as relevant to navigation, what might be the advantage of the Lindemann irrational pi over that of, say, 355/113?

[C1ay]

Then tell us, Clay, as relevant to navigation, what might be the advantage of the Lindemann irrational pi over that of, say, 355/113?

Accuracy. Of course now you want to taunt with a much better approximation than that poor value of 3.1640625 you keep wanting to proffer as a finite pi value. Someone using that value across the universe could find themselves off course by light-years over relatively small galactic distances.

[Robust]

Buffy, You give quite a bit to chew on.and doubt that I respond in kind. Let me just point to the crux of the problem I have with the Lindemann irrational pi. It is not because it is irrational oer say that is my objection to it, but that it is clained and taught to be sacrosanct. It clearly is not! I also strongly objectt to the notion that th area to any closed continuum can be defined by other than a whole number or ending decimal - which the Lindermann pi does not allow.

[Buffy]

I also strongly objectt to the notion that th area to any closed continuum can be defined by other than a whole number or ending decimal - which the Lindermann pi does not allow.

Well, that pretty much explains it. No mathematician would agree with you at all, but a bunch of us here would love to understand *why* you think that the area/radius ratio of a circle (and by the way many other mathematical solutions), *must* have a representation in a finite number of decimal places... Like I say, this issue has not been debated for millenia, so its a surprising conjecture at the very least.

 

Cheers,

Buffy

[Robust]

Well, that pretty much explains it. No mathematician would agree with you at all, but a bunch of us here would love to understand *why* you think that the area/radius ratio of a circle (and by the way many other mathematical solutions), *must* have a representation in a finite number of decimal places... Like I say, this issue has not been debated for millenia, so its a surprising conjecture at the very least.

 

Cheers,

Buffy

Hi, Buffy! I truly enjoy your input! The way I see it is that there 2 sides to this coin - and in dire need of a devil's advocate. We are making strident and truly remarkable advancement in the applied sciences. On the other side of the coin is the academia, which I see as still mucking about in the Dark Ages - as has always been the case. No offense intended should you belong to the latter - just the way it has always been, and perhaps out of necessity. I'm in the fortunate position of no longer belonging to either - so able to play both ends. It's how the Forum came about iat the start of it all....if I read the history of it correctly.

[Qfwfq]

I also strongly objectt to the notion that th area to any closed continuum can be defined by other than a whole number or ending decimal - which the Lindermann pi does not allow.

Why do you say the Lindermann pi does not allow it?

 

Why can't a closed continuum have an irrational area? You agreed that root two is irrational, after I had shown that root two over six is the area of a closed continuum.

[Robust]

Why do you say the Lindermann pi does not allow it?

 

Why can't a closed continuum have an irrational area? You agreed that root two is irrational, after I had shown that root two over six is the area of a closed continuum.

I'm not saying that the closed continuum cannot have an irrational area. I'm saying that the closed continuum must be described by a whole number or ending decimal - which the Lindemann pi does not allow.