Dear Scholar

Basically, square and circle are SAME.

Differences DOES exist.

For example : diamond and coal do appear different

But, basically both are made of the element CARBON

Dear Scholar

We have 2 formulas as proof

So, I am not making problem

Dear Scholar

Present polygonal number as pi of the circle is an approximation ,because 3.14159256.....is in a decimal form

In the two formulas which are revealed by the Nature is in surd form

(14-root2)/4.

root2* root2 = 2

So , a surd is an EXACT number. And its decimal form is always an APPROXIMATION

2. The 2 formulae are based on RADIUS of the circle

3. The present pi number 3.14159256.....is IMPOSSIBLE to adjust itself with the RADIUS of the circle.

4.Even Archimedean 22/7 is also incompatible with RADIUS of circle.

5. For 2200 years we have been UNFORTUNATELY following an alien number 3.14159256...as Pi of the circle.

6. When the real pi number of CIRCLE based on its RADIUS is available since March 1998 ,we have been ignoring it.

NATURE will never pardon us if the real pi is ignored.

I learn and share what I believe well substantiated

OK my Dear Scholar

You have every right to criticize my wprk

Thank you

it is NOT a threat

as you please

Dear Scholar

Here is a truth for your perusal

Dear Scholar

You say n*delta is insignificant. OK

You must remember that in Exhaustion method of Archimedes at every doubling of sides of polygon we invariably use irrational number root3.

However , we take only just 10 decimals of root3 and we LEAVE INNUMERABLE DECIMALS to the winds. It is called round up or truncation.

At every step we are taking a wrong value and at the end ,say ,10000 sides of the polygon, we get perimeter/ area of it. It ( 3.1415926...) is definitely a lower value.

So. we are attributing a wrong number of polygon to circle as its pi.

2.

What is Pi = circumference/ diameter

What are we doing here = perimeter of the polygon/ diameter of the circle

3.Diameter is 1

4. Polygon has 2 lengths passing through center

one length is less which connects mid points of 2 opposite sides of the polygon

5. So, numerator is LOWER VALUE/ denominator is 1. The denominator should be LESS than DIAMETER

My dear scholar

If you say there is NO truncation I have nothing to say.

You can contact Prof. Kanada of Tokyo University , Dept. of Computer , because he was the person who used Super Computer using the infinite series of Prof. Simon Plouffe et.al

OK my scholar

Well, of course, but I have to tell you that the Bible (Old Testament) has priority over your invention. It is beautifully stated there that the circumference of a round temple is 3D, where D is the diameter. No pi involved whatsoever and no radicals, nothing. Very sexy!!! ;)

Dear Scholar

Yes , you are RIGHT.

Holy Bible's Pi equal to 3 is the FIRST.

If we take you exact answer for PI =[14-Root 2]/4, it will miss the actual value to the 15th decimal point by

0.004853955816933

PI is endless and the error can not be simply dismissed by assuming your answer is exact. Amir Al-Khafaji

Dear Scholar

(14-root2)/4 is EXACT value

In decimal form it is NOT exact . 3.1464466................... no ending

2. We have a circle having exact area equal to that of square and is done by Prof. Gernot Hoffmann of Embden, Germany.

Here is my book attached and you can see in page number 163 , CIRCLE A SQUARE

my dear scholar,

Try the ratio of 355/113 because it will give you a much better approximation for PI than (14-root2)/4!

Amir Al-Khafaji

"2. We have a circle having exact area equal to that of square and is done by Prof. Gernot Hoffmann of Emden, Germany." -- You are mistaken, my dear Sarva Jagannadha Reddy: Gernot Hoffmann clearly says (in section "5. Method C") that "Neither for ‘circling a square’ nor for ‘squaring a circle’ exists an accurate solution by ruler and compass." Also, he states that your approach "is an approximation as well", and he quantifies the accuracy of this approximation by using the standard pi value (3.1415926...) as reference for the circle area.

Dear Scholar

If Prof Gernot Hofmann says Cosmic Pi is right and real and Exact he would be "out" before the accusing voices of the world

It happened to many people who supported Cosmic Pi in the past.

For that reason I am ALONE facing the drama of attack by the supporters of 3.141...

"If Prof Gernot Hofmann says Cosmic Pi is right and real and Exact he would be "out" before the accusing voices of the world" -- No: Anybody who says Cosmic Pi is right and exact is "out", however not "before the accusing voices of the world" but just because of the pure mathematics itself.

Dear Scholar

Better, I advise you forget Cosmic Pi and sleep peacefully .

Why should you break your head.

I told you already why I do alll this: On the one hand, I want to avoid that people start to mistrust science (I've discussed that elsewhere). On the other hand (and that is a new aspect), I have become curious about the true meaning behind (14 – sqrt2)/4 and all your geometrical constructions. And if there is something deeper behind, you are the person who should receive the honour for it, so please don't throw that chance away by stopping to discuss about "Cosmic Pi" here on ResearchGate.

Of course, as Gernot Hoffmann stated, as a purely geometrical construction -- as opposed to other approximative values like 355/113 -- your value (14 – sqrt2)/4 is a rather good approximation and delivers the best solution so far. (Therefore, this would be a real challenge you could present the world of mathematics -- if somebody would find an even better approximation just from a purely geometrical construction.) But I guess the fact that (14 – sqrt2)/4 is a rather good approximation is only a side product; as I said above, probably there's more to it.

J.-M. Wagner: "This would be a real challenge you could present the world of mathematics -- if somebody would find an even better approximation just from a purely geometrical construction."

Well, it seems that such geometrical constructions have already been found -- namely by Ramanujan, published in 1914: "Approximate geometrical constructions for Pi." Quarterly Journal of Mathematics 45 (1914): 350--374. So, I think it's really worthwhile that you try to get a look at this paper, my dear Sarva Jagannadha Reddy. Do you have any idea whom you could ask?

Dear Scholar

I did nothing for Cosmic Pi

Gernot Hofmann's costruction in Circling a square gives the AREA of the circle as 4, equal to that of the AREA of the square :4, whose side is 2.

So, it is EXACT and NOT approximation

Professor has also said " a cute method how to establish pi' by elementary geometrical constructions"

"I did nothing for Cosmic Pi" -- Well, you (and only you) posted a lot of "Cosmic Pi" stuff here on ResearchGate.

"Gernot Hofmann's construction in Circling a square gives the AREA of the circle as 4" -- No: He explicitly writes "Square-Area/Circle-Area = 1.001545", i.e. the area of the circle is somewhat smaller than 4.

"Professor has also said a cute method how to establish pi' by elementary geometrical constructions" -- Yes, indeed; but note that he wrote pi with an apostrophe! And this is completely on purpose, because he also wrote (as first remark under "6. Method D"): "This method by R.S.J.Reddy [4] is an approximation as well. The author wants to replace the common value for π by an algebraic irrational expression π´." So, he definitely says that "Cosmic Pi" is an approximation.

Dear Scholar

1.I am instrumental in the hands of NATURE. I consider myself a SCRIBE . OK

2. We obtained a line segment to the required RADIUS which gives area of the circle 4 with COSMIC PI VALUE. Please go through his PROOF once again. Prof Gernot Hofmann is NOT NOT wrong.

3. I told you already, now this period from March 1998 is a TRANSITORY PERIOD wherein both Pi values 3.141.... and 3.146... are kept side by side and discussed as Pi and Pi'

4. The meaning for the word approximation is ALSO 'estimation'

So, to conclude , area 4 is EXACT and approximation means estimation.OK

"1. I am instrumental in the hands of NATURE. I consider myself a SCRIBE. OK" -- No, this is not OK, because intuition is not enough! When you don't check that there are no contradictions, you cannot claim that you've found something true! Remember what Isaac Newton put an emphasis upon: "I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy."

And William Whewell stated: "What is requisite is, that the hypotheses should be close to the facts, and not connected with them by other arbitrary and untried facts; and that the philosopher should be ready to resign it as soon as the facts refuse to confirm it." (https://en.wikipedia.org/wiki/Hypotheses_non_fingo)

"2. We obtained a line segment to the required RADIUS which gives area of the circle 4 with COSMIC PI VALUE. Please go through his PROOF once again. Prof Gernot Hofmann is NOT NOT wrong." -- Indeed, he isn't wrong, because right from the beginning he says (please read this yourself again): "This method by R.S.J.Reddy [4] is an approximation as well. The author wants to replace the common value for π by an algebraic irrational expression π´. π = 3.141592...; π′ = 3.146446... [...] This is a good approximation" You see? Two times he says approximation, which means that it is not exact.

Gernot Hofmann explicitly quantifies how much the approximation deviates from the true value: "Square-Area/Circle-Area = 1.001545", and since the square has an area = 4, the circle has an area = 3.993829..., which is smaller than 4. At the end, he writes explicitly: "r3 = sqrt[16 / (14 – sqrt2)]" (as a result for the radius obtained by the Cosmic Pi construction) and "true radius: r = 2 sqrt(1/pi)" (with the unprimed pi). And since Cosmic Pi is larger than pi, the true radius is larger then the approximmative value using Cosmic Pi.

You can see this also from the drawings in the PDF file of Gernot Hofmann: In all the methods presented there, the red circle is the result of the respective geometrical construction, whereas the black dashed circle is the true one (this can be seen most clearly on page 2 for method A).

"3. I told you already, now this period from March 1998 is a TRANSITORY PERIOD wherein both Pi values 3.141.... and 3.146... are kept side by side and discussed as Pi and Pi'." -- This doesn't make sense because a circle has a well-defined single value for his circumference when the diameter is given, namely circumference = pi times diameter. Therefore, only one true value can exist for pi; all other values are wrong. There are many proofs that pi = 3.141592... is the true value (not only the polygon approach), but you refuse to see the arguments.

For example, to come back to the polygon approach because it's so simple and geometrically clear, have a look at this: http://de.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/29504/versions/23/previews/html/ComputationOfPiByArchimedes.html?access_key=#1

"4. The meaning for the word approximation is ALSO 'estimation'." -- Yes, and this also means 'not exact'! So, to conclude, area = 4 is exact for the square only, but not for the circle.

Dear Scholar

There is no SINGLE proof to call 3.141... as Pi value.

There are mathematicians who propose 3.143.., 3.144.. as pi value.

Be happy you believe there is a proof.

OK

"There is no SINGLE proof to call 3.141... as Pi value." -- Well, I told you already that here in our discussion I don't care about the true pi value; I just put "Cosmic Pi" to the test. And so far, it hasn't passed the test successfully, because you are reluctant to perform it appropriately.

"There are mathematicians who propose 3.143.., 3.144.. as pi value." -- This is just sidestepping; here, we're not discussing these values but "Cosmic Pi".

"Be happy you believe there is a proof." -- Have a look around you, and you will find many proofs! I'll show you some of them later.

Dear Scholar

Basically 3.141.... is polygon's number

When a recipient gets kidney from some body on transplantation it does NOT become his REAL kidney

"Basically 3.141.... is polygon's number" -- Well: Show me the polygon that has exactly this number as its circumference length!

Dear Scholar

That is what I am saying all these 20 years.

Why should we borrow 3.141.. of polygon and make it or thrust on circle as Pi ?

Do you agree if I take your shirt because it fits me well as mine ?

2. If 3.141... is really a Pi number get this value with radius alone while calculating area of the circle as is done with Cosmic Pi in the attachment.

ACCEPT THIS CHALLENGE please

OR forget COSMIC PI immediately.

Stop: You are the one who claims that 3.141... is the length of a polygon. But I doubt that, so I challenge you to show me the relevant polygon.

And with regard to your challenge to calculate the area of the circle using its radius alone: I thought this had already been done by Ramanujan, "Approximate geometrical constructions for Pi"; however, it seems that the respective reference (including the title) is completely wrong. However, in the literature there is lots of stuff on pi calculations, so I'll keep digging; I'll let you know as soon as I find something relevant.

Dear Scholar

Here is the Table for 3.141... of a polygon

2. Ramanujan never derived pi value equal to 3.141... from a geometrical construction.

3. He DID square a circle up to few decimals of 3.141... giving a geometrical construction.

4. " All stuff" do not use construction. All stuff used " INFINITE SERIES "

"Here is the Table for 3.141... of a polygon" -- None of the values in this table is identical to standard pi = 3.141592653589793238462643383... So, I challenge you again to show me the corresponding polygon.

"2. Ramanujan never derived pi value equal to 3.141... from a geometrical construction. 3. He DID square a circle up to few decimals of 3.141... giving a geometrical construction." -- So, what does that mean for "Cosmic Pi"?

"4. 'All stuff' do not use construction. All stuff used 'INFINITE SERIES' " -- This is not fully true, because the infinite series is based on single steps with each of them being based on a geometrical construction.

Dear Scholar

1. There is no polygon construction for many decimals of 3.14159265358...

2. To show that Ramanujan depended on other than constructions for the 3.141.... which was in vogue then.

4. It is FULLY true.

5. There is NO SECOND GEOMETRICAL CONSTRUCTION for 3.141....

--------------------------------------------------------------------------------------------------

6. 3.141.... is TOTALLY INCOMPATIBLE with RADIUS of the circle

This reason is enough to assert and DISPROVE that 3.141.....is NOT Pi number

Surely this looks like a more interesting conversation than I would have predicted ;)

Dear Scholar

Thank you

"1. There is no polygon construction for many decimals of 3.14159265358..." -- So, how can you then say that 3.14159265358... is a polygon number? And if there is no polygon construction for all those decimals, where do they come from?

"2. To show that Ramanujan depended on other than constructions for the 3.141.... which was in vogue then." & "3. He DID square a circle up to few decimals of 3.141... giving a geometrical construction." -- So, again, what does that mean for "Cosmic Pi"?

"5. There is NO SECOND GEOMETRICAL CONSTRUCTION for 3.141..." -- Sorry, but this is wrong: Take your Big Wheel, roll it over the ground, and measrue the circumference length as straight distance on the ground (cf. https://de.wikipedia.org/wiki/Kreiszahl#/media/File:Pi-unrolled-720.gif). Isn't that a geometrical construction, too?

"6. 3.141.... is TOTALLY INCOMPATIBLE with RADIUS of the circle" -- How do you know without having tried to roll your Big Wheel?

And back to our circumscribing polygons (only a few of them, not an infinite series!): Isn't that a geometrical construction, too?

Dear Scholar

1.There is NO polygon construction. But the entire process is just by CALCULATION only and done theoretically up to 10000 sides of the polygon. They have come by CALCULATION.

2. Cosmic Pi is WHOLLY backed by geometrical constructions.

3. If you know any second geometrical construction other than of EXHAUSTION METHOD , let me know please!

4. Show me any SINGLE method either geometrically and or INFINITE SERIES wherein RADIUS plays a significant role in obtaining 3.141....

5. Circumscribed polygon has NO NO NO NO NO NO..............RADIUS in it. Radius and Circle are DIVINE and very basic entities. ALL Stars and Planets are as old as our Physical Universe of 13 to 14 Billion years.

"1.There is NO polygon construction. But the entire process is just by CALCULATION only and done theoretically up to 10000 sides of the polygon. They have come by CALCULATION." -- Yes, of course: Numbers are obtained from calculations, aren't they? Also "Cosmic Pi" comes from calculations, both related to the geometric construction itself, but also

"2. Cosmic Pi is WHOLLY backed by geometrical constructions." -- So is standard pi: circumference / diameter.

"3. If you know any second geometrical construction other than of EXHAUSTION METHOD, let me know please!" -- This is an interesting question, and I will try to make some suggestions.

First suggestion (as above): Roll your Big Wheel over the ground and measure the circumference length as straight distance on the ground.

Second suggestion: Do the inverse on a computer, by drawing a circle with a certain radius and wrapping a perfect measuring tape (of thickness zero) with tick marks around it, then read the circumference length at the point where the full loop is finished.

Third suggestion (approximative value only): Divide a circle made of paper or cardboard into an even number (at least 12) of equally sized wedges, cut them out, symmetrically split one of the wedges in two half-sized sectors, and rearrange them side-by-side to form a nearly rectangular pattern (with the two half-sized sectors at the ends); the height of this quasi-rectagle is the radius of the circle, and the length is half the circumference, i.e. pi times the radius. (This method is illustrated, for example, here: http://www.mathsisfun.com/activity/pi-approximation.html)

More suggestions: to be done later.

"4. Show me any SINGLE method either geometrically and or INFINITE SERIES wherein RADIUS plays a significant role in obtaining 3.141..." -- See suggestion #3 above; for finer and finer division of the circle in more and more wedges, the closer one comes to pi.

"5. Circumscribed polygon has NO NO NO NO NO NO..............RADIUS in it." -- Sorry, but this is wrong: A regular circumscribing polygon has an incircle, and this, of course, has a radius; this radius is the distance from each of the segments of the polygon to its center. (For an illustration of the general case, see https://en.wikipedia.org/wiki/Tangential_polygon)

Dear Scholar

1st Suggestion: Your suggestion is already adopted in the 118th method :attachment

2nd Suggestion; I have been asking computer specialists to measure the time taken for CURSOR to travel on the perimeter of the superscribed square and the time taken for CURSOR to travel on the circumference.

Time taken for square/ time taken for circle.

Compare the result with that of 3.141... and Cosmic Pi 3.146.....on denominator and 4 on the numerator

3rd Suggestion: a crude method

4. Here is lies a blunder

We take the perimeter of the CP and divide it with diameter of the inscribed circle.

It has 2 clear ERRORS

1. We should divide perimeter of the polygon with the chord that passes at the center but MEETING at the MID-POINTS OF TWO OPPOSITE SIDES of the CP

This length is NOT diameter but it should be at the denometer

How?

Perimeter of CP/ MID-points of the sides of the polygon.

WITH THIS WE GET Pi as equal to MORE than 3.141.....

--------------------------------------------------------------------------------

2nd error:

Perimeter of the CP/ Diameter of the circle = 3.141.... is WRONG

How?

Definition of Pi = Perimeter of the circle/diameter of the circle

---------------------------------------------------------------------------

Thus, 3.141.....is a HYBRID PI VALUE because polygon on numerator and circle on denominator.

THUS for the last 2200 years from Archimedes we have been using the hybrid pi value as PURE PI VALUE.

"1st Suggestion: Your suggestion is already adopted in the 118th method :attachment" -- Yes, the principle behind this method is described there, but the measurement itself, reaching an accuracy of 0.13 %, is not presented in "118th method.pdf": Where are the actual numbers for the diameter and the "rolled-over" length? Where are the actual numbers for the precision of the measurement method? Where is the consideration for the total (accumulated) error of the final result? Without all these data, you haven't proven anything. Do it; only then we really know what we're talking about.

For example: In your video "Exact Pi-by experiment.mp4" you measure the following data: circumference = 295.0 mm (exact up to plus/minus 0.5 mm; at position 7'14" in the video) and chord length = 67.0 mm (exact up to plus/minus 0.5 mm; at position 8'16" in the video). Unfortunately, you did not measure the diameter directly, so one has to calculate it as sqrt2 times chord length, giving diameter = 94.8 mm (exact up to plus/minus 0.7 mm).

Therefore, the ratio is 295.0 mm / 94.8 mm = 3.11181..., where the position of the last relevant digit follows from the accumulated error as follows: relative error of circumference = 0.5 mm / 295.0 mm = 0.0017 = 0.17 %, relative error of diameter = 0.7 mm / 94.8 mm = 0.0074 = 0.74 %, relative error of ratio = sum of relative errors = 0.91 %, so the absolute error of the ratio is 0.028, leading to the final result of 3.112 ± 0.028. How can you say that this proves Cosmic Pi?

Dear Scholar

Your approach does NOT serve the purpose of finding the EXACT PI VALUE called Cosmic Pi

You may please see the following and understand the aim of the experiment

VERY IMPORTANT MESSAGE AFTER SEEING THE EXPERIMENT This experiment has 2 parts. In the 1st part we measure their dimensions with ruler. They are NOT accurate.As far as ruler values are concerned, though approximation ,but agree with each other. As they are right ,we go to the 2nd part. The answer is to VERIFY the EQUATION. When Equation is proved Right by the ruler values then we have to go to the equation and DERIVE the REAL Pi Value. So, Cosmic Pi is NOT NOT NOT .....derived from the experimental values . On the other hand, Pi is derived from the VERIFIED EQUATION . Hence it is the NATURAL Pi Value

OK

"Your approach does NOT serve the purpose of finding the EXACT PI VALUE called Cosmic Pi" -- Wait: This doesn't work at all! In research, one never gets what one wishes, but what one finds out. And in order to really find something out, one has to be fully open-minded, which especially means that one must not be biased by one's own expectations and/or pre-knowledge of the situation. And one has to be fully aware of the following:

In one's mind one has to make a clear separation between theory and experiment, because what happens in the real world is completely independent from the theories we make about the real world; whether we have our theories in mind or not, or whether we write them down or not, doesn't change reality, and whether our theory is right or wrong, the real world events don't care -- they are just as they are, and we have find out how they really are.

So, here we are about to just find out the value of pi according to its definition as the ratio between circumference and diameter. We both have our theories about it, but without an experimental test a theory is worth nothing, absolutely nothing (!!) -- at least in natural sciences.

Note: The point is that, because of real world and theory being independent of each other, by such an experiment all theories are put to the test, including "Cosmic Pi"! Without a "reality check", there is no reality in any theory about the real world. And note also that if there is a discrepancy between theory and experiment, the experiment wins! (For me as an enthusiastic theoretician this was a very hard lesson to learn, but it's worth it.)

I have asked you beforehand whether such an experiment would help us to clarify whether 3.146 or 3.142 is a better approximation for the ratio between circumference and diameter, and you have agreed. So, from now on there is no more sidestepping; we just have to look sharply at our experiments and their outcome.

In your video "Exact Pi-by experiment.mp4" you reach an accuracy of about 0.9 %, and this is not sufficient to distinguish between 3.146 and 3.142, because they differ by 0.13 % only. This means that your measurement was still too coarse; we need about one order of magnitude better precision in order to come to a valid conclusion. In other words: So far, the question about "3.146 or 3.142" is still open.

However, luckily we've already learned something important from the measurement in your video "Exact Pi-by experiment.mp4": There, the largest contribution to the relative error comes from the measurement of the diameter. Since the positioning and read-out errors are always the same, we can reduce the relative error of the diameter measurement by using a significantly larger circle. You have such a large circle, it has a diameter of about 1750 mm, which by assuming an absolute error of 1 mm gives a relative error of only 1 / 1750, which is 0.057 %. So, we have a good chance that this experiment works.

Dear Scholar

Thank you