Irrational Pi Defrocked

[Robust]

I do understand that if you start from a given area with a given pi value, you come to a given radius. If you use that given radius, with the given pi value, you get back to the given area. That is just the nature of equations.

Problem is, both area and radius are connected in a circle. It is impossible to get two circles with the same area, yet different radii. This implies that pi is a constant, and not an arbitrary value.

I truly do not understand the notion you have of there being 2 circles to the equation. There are different values to the radius only because of the use of different pi values. I do think i'm beginning to see your confusion on this . Give me a bit of time to think of how I might clear it up..

[Qfwfq]

I truly do not understand the notion you have of there being 2 circles to the equation.

True Robust, because indeed the two couldn't both be circles! In Euclid's plane, only one circle can have a given area and it will have one value of radius. The ratio will be given by the accademically ingrained, rational and transcendental value of pi.

 

To get a different "pi" you need to define your circle on the surface of a semi-cone. This way you can have pi*(sin alpha) instead of pi. To have a value greater than the accademically ingrained, rational and transcendental one you can use a 2-D manifold analogous to the conic surface that can be embedded in flat 3-D with radial furls. Even on such surfaces, the ratio of circumference to radius depends on where the circle's centre is and it becomes exactly 2pi if the vertex isn't internal to the circle.

 

Is that tree pruning, or constructive criticism?

[MortenS]

I truly do not understand the notion you have of there being 2 circles to the equation. There are different values to the radius only because of the use of different pi values. I do think i'm beginning to see your confusion on this . Give me a bit of time to think of how I might clear it up..

 

So, this means that it is wrong to use different pi values, which again means that it is not arbitary what pi value you mean, which means pi is a constant. I feel we are coming some way here. If we can agree that the pi value is a constant, we can debate wether it is rational or irrational, and if irrational, whether it is transcendent.

[C1ay]

I truly do not understand the notion you have of there being 2 circles to the equation. There are different values to the radius only because of the use of different pi values. I do think i'm beginning to see your confusion on this . Give me a bit of time to think of how I might clear it up..

Yeah, you're right, there can only be one circle. How about we start with a circle that has an area of 16. Show us how to get the exact diameter of that circle using several of your pi values. That's where the confusion lies. A circle only has one diameter, how do you find that one diameter with any pi value you want to pick out of thin air. If I want to turn a cirlce on my lathe that has an area of 16, what diameter do I need to cut?

[Rincewind]

The new formula was given as solving a mechanical engineering problem requiring a given radius for the turning of a circular disc of any specified area

 

...

 

For my sake I was hoping we might keep this topic simple as possible - my not being a maths person persay.

 

...

 

sqrt area/sqrt 3.14159265359 pi = 2.25675833419 radius; r^ pi = 16 area;

sqrt area/sqrt 3.14159292035 pi = 2.25675823838 radius; r^ pi = 16 area;

sqrt area/sqrt 3.16049382716 pi = 2.25 radius; r^ pi = 16 area;

sqrt area/sqrt 3.1640625 pi = 2.24873078056 radius; r^ pi = 16 area.

 

...

 

Sqrt 16 = 4; sqrt pi = 1.7724....; 4/sqrt pi = 2.2567; 2.2567....squared = 5.0929 radius; r^*pi = 16 area.

 

...

 

I truly do not understand the notion you have of there being 2 circles to the equation.

 

Let me try to explain why it appears you have multiple circles to the equation.

 

You are starting with a fixed area to your circle, then saying that you can derive the radius to give to an engineer to cut a circle of fixed area 16 using any value for pi. As you can see from the above equations showing your working out, each different value of pi gives you a different value for the radius:

 

Using pi value of 3.14159265359 you get 2.25675833419;

using pi value of 3.14159292035 you get 2.25675823838;

using pi value of 3.16049382716 you get 2.25;

using pi value of 3.1640625 you get 2.24873078056.

 

Which of these values for the radius are you going to give the engineer so that he or she cuts you a circle of area 16? Or are you suggesting that it doesn't matter which of these radius values you give the engineer, the resulting circle will be 16 units in area?

[Qfwfq]

Or are you suggesting that it doesn't matter which of these radius values you give the engineer, the resulting circle will be 16 units in area?

Yes, that is what Robust claims.

 

The simple reason for his claim is that, if you use the same value of pi to calculate the area, you will recover the same initial value. Robust takes this tautological fact as a confirmation of his claim.

[Robust]

Yes, that is what Robust claims.

 

The simple reason for his claim is that, if you use the same value of pi to calculate the area, you will recover the same initial value. Robust takes this tautological fact as a confirmation of his claim.

The actual cutting of the disc is irrelevant - but a hypothetical question showing that pi is arbitrary and that it is actually root 2 that rules, perhaps the best example being the trianglature formulae giving distance on the quadrant arc and its accompaning chord length. (Which can be given if you wish.)

[C1ay]

The actual cutting of the disc is irrelevant

No it's not. I want to construct a hydraulic scale that I can use to weigh things. To build this scale I need a cylinder and a piston that have exactly one square inch of area. I can then use a pressure gauge calibrated in pounds per square inch to read off the weight directly. What diameter will my cylinder and piston need to be?

[Robust]

No it's not. I want to construct a hydraulic scale that I can use to weigh things. To build this scale I need a cylinder and a piston that have exactly one square inch of area. I can then use a pressure gauge calibrated in pounds per square inch to read off the weight directly. What diameter will my cylinder and piston need to be?

 

Sorry, can't help you with that one. The engineering problem was purely hypothetical. Let me simplify by merely asking what radius gives an area of 16 to a circular plane?

[C1ay]

Sorry, can't help you with that one. The engineering problem was purely hypothetical. Let me simplify by merely asking what radius gives an area of 16 to a circular plane?

Can't help? Just calculate the correct radius. I can double it to get the diameter.

[maddog]

Clay, tthe mathematical proof is given by the formula. It is not conjecture:

 

sqrt area/sqrt pi = radius;

r^2*pi = area.

 

You cannot hide from it....use any known pi value and you will get the same area.

I quite agree with C1ay. If I were to have a formula like yours

 

pi * r^2 = area

 

Let area = 1 here then

 

pi = 1/r^2

 

If r were held fixed (any fixed value) how could pi vary ("any know pi value").

Now let r = 1 => pi = 1 (this is a know value of pi by your definition -- albeit a very

bad one)... :xx:

 

If x * pi = 1 => pi = 1/x Here x is a dependent variable of pi (pi depend on x).

 

There is only one known value for pi (the ratio of area to radius squared) which

just happens to the same as the ratio of the Circumfrence by the diameter. Both

are the same. What you might be referring to are all the approximations of pi

of which there are lots. These approximations tend towards the value for pi that

is given in this thread (of which is also an approximation). So what is your point... :xx:

 

Maddog

[Rincewind]

Let me simplify by merely asking what radius gives an area of 16 to a circular plane?

Well, Robust, you have given us four different values for that radius:* 2.25675833419, 2.25675823838, 2.25 and 2.24873078056.* Why don't you save us all some anguish and tell us which one is the correct one in your opinion?

[C1ay]

Introduced on an Australian science forum is a formula showing Euler's highly touted irrational pi to have no greater authority than any of the other numerous pi values of historical reference. The new formula was given as solving a mechanical engineering problem requiring a given radius for the turning of a circular disc of any specified area - given as follows:

 

FORMULA: sqrt area/ sqrt pi ^ = radius; thus, radius * pi = area.

 

Please note change in formula from that originally given.

The formula gives the radius to any circle of a prescribed area - regardless of any and all known pi values as might be applied. What do you think?

Just wondering, why didn't you make the same claims here at physicsforums that you've made here? Why don't you share these claims with them as well?

Use any other known pi value in place of the irrational pi and you will derive the same answer. The irrational pi is not sacrosanct.

The earliest known pi value we have dates some 5,000BC - 256/81 - an engraving in stone. There are a number of others of historical record: 355/113; 22/7, etc. I tend toward the finite value of 3.1640625 derived from my own studies - but the book is not closed on that as yet.

Clay, you are correct in that I am not a maths person persay, but mistaken on all other counts. As regards the rasius of a circle, it is whatever the circumference dictates, pi being nothing more than the ratio of line to arc, radius describes the line and radian the arc it subtends. Pi is but the ratio of one to the other, not the determinasnt.

 

History records a number of authenticated oi values. There has been another since the irrational pi - a finite pi value of 3.1640625. All that notwithstanding, the formula given here proves that one pi value is no more outstanding than another as to describing area of the circle.

The actual cutting of the disc is irrelevant - but a hypothetical question showing that pi is arbitrary and that it is actually root 2 that rules, perhaps the best example being the trianglature formulae giving distance on the quadrant arc and its accompaning chord length. (Which can be given if you wish.)

Especially since you've remarked over there:

I took a bit of abuse by contempories for this formula - wanted to hear what Y'all might think of it.

and

But the implications of this one does present a serious conflict as regards the pi value, showing it to be irrelevant. the radius is given consistently regardless of the pi pi value employed

and

It's a hypothetical question. the thickness and other parameters are immaterial. Only the radius to the circular plane is required.

and best of all...

Knitpicking is fine with me - probably the more the better!

BTW, just a small nitpick, it's nitpicking, not knitpicking, brewnog had it spelled right to begin with.

 

So why don't you just invite brewnog, pack_rat2 and FredGarvin from physicsforums to join in over here since more nitpicking is better? It will also save you from having to login into 2 different places to get nitpicked :xx:

[Robust]

Well enough Clay. Let's leave off the "nitpicking" then and cut to the chase - which you seem to want to avoid. Would you agree then that the area to any closed continuum must be described by a whole number or endeing decimal?

[C1ay]

Well enough Clay. Let's leave off the "nitpicking" then and cut to the chase - which you seem to want to avoid.

Let's do cut to the chase. You said,

 

The formula gives the radius to any circle of a prescribed area - regardless of any and all known pi values as might be applied.

Now support your claim. It is blatantly clear to any maths person that a circle with a given area has one, and only one, radius. There is only one value of pi that will give that radius. Pi is not a variable that you can adjust at will.

 

Would you agree then that the area to any closed continuum must be described by a whole number or endeing decimal?

No! It has nothing to do with proving your claim anyhow. Calculate the area of a circle with radius 1 as an example of a finite area with an irrational measure.

 

BTW, did you invite your other nitpickers to join us?

[Robust]

Let's do cut to the chase. You said,

 

 

Now support your claim. It is blatantly clear to any maths person that a circle with a given area has one, and only one, radius. There is only one value of pi that will give that radius. Pi is not a variable that you can adjust at will.

 

 

No! It has nothing to do with proving your claim anyhow. Calculate the area of a circle with radius 1 as an example of a finite area with an irrational measure.

 

BTW, did you invite your other nitpickers to join us?

 

Works the same way, Clay:

1/sqrt pi = radius; r^2*pi = 1 area.

As i said, it is root 2 that rules - not pi.

[C1ay]

Works the same way, Clay:

1/sqrt pi = radius; r^2*pi = 1 area.

As i said, it is root 2 that rules - not pi.

Not area = 1, radius = 1.