I don't understand step #4: Why does it hold that CH = (2 - sqrt{2}) / 4 = \pi - 3? Please elaborate on that.
Well, this doesn't help, since it just uses the same relation without justification (step #6). So, please provide an independent argument that shows the correctness of this relation.
Sorry, this is all way too lengthy compared to the simple question I have. At least from the second file (VOL_VI_Cont...2.docx) I was able to understand that CH = (2 - sqrt{2}) / 4 (cf. page 2: Part I, step #11). However, it's not clear why this further equals \pi - 3. So, please point out (in these documents) where you elaborate on that.
My Dear Scholar
After a study & search from 1972 to 1998 I could understand INTUITIVELY that CH = Pi-3.
Then from 1998 to today I have CONFIRMED CH = Pi-3.
Here is one example to ask a question
HOW all spherical Celestial Bodies float and rotate "in Space" ( wrong but I have to use here "in" ) ?
The question is simple
The answer is we SEE the heavenly bodies and accept it the REALITY
And not ask WHY, HOW, for WHAT etc.
In this work on COSMIC Pi we can not disprove it
But asking a stereotyped question like proof, justification we are delaying to use the REAL/ TRUE & EXACT Pi.
Ok
Regards
Well, you say that you have "CONFIRMED [that] CH = Pi-3." What do you mean here by "CONFIRMED"? And, at the same time, what do you mean in your starting question by "Why"?
Dear Scholar
The word "confirmed " I mean everything all these 19 years 184 different constructions ( methods ) have supported CH = Pi -3 , directly and/ or indirectly leading to only Cosmic Pi = (14-root2)/4=3.1464466....
Further, there are a few tests which support Cosmic Pi and are submitted here
I may be excused many papers are dumped on you !
First of all: Happy new year!
Well, I have to apologize, because what you answered is not what I had in mind -- I should have put my question more clearly. My question about "confirmed" and the "why" was intended to rather refer to the method of confirming and providing a valid reason. So, I think that here we use formal mathematical logic. Do you agree, or do you seek a reason in another sense?
My Dear Scholar
Thank you
" Method of confirming and providing a valid reason "
I mean it is different
How?
You may expect confirmation by adopting well KNOWN class room education or described in a related fields explained in a book .
The new pi is well BEYOND all these known methods and concepts
ALL 184 methods adopted in confirmation of the 1998 pi value is TOTALLY DIFFERENT AND ORIGINAL AND NO WHERE WE FIND THEM in any book.
You walk in HIGH ROAD
But I went in WOODS deviating my path LEAVING HIGH ROAD , no body entered before
And that is the reason why people of indoctrinated education are RELUCTANT to accept this new pi called Cosmic Pi
And another thing is so far no body could DISPROVE it
I mean the present position about this REAL Pi is
RELUCTANCE TO ACCEPT
&
UNABLE TO DISPROVE IT
OK
So, do we talk about science or about religion? (This is just about the relevant premises and methods; for me there is no perfectly strict separation between these two -- well, except for certain personal aspects regarding "truth" and "reality".)
Dear Scholar
Discoveries are original concepts and that is science.
Science is a self-correcting one
Religion has nothing to do here
I am very sorry
"Science is a self-correcting one" -- I agree! But how does it work? What is the reference by which one can decide about necessary corrections? Here is my suggestion: If something contradicts basic facts, there are two possibilitites: either it is wrong and has to be discarded, or the basic facts do not hold. Do you agree?
Interesting academic debate Dr. Sarva and Dr. Jan-Martin. I am following and learning.
Dickson
"Let us close please !" -- Why? Things aren't settled yet: Please clarify the contradiction that on the one hand, you want to do science, but on the other hand, your most important step in all of your works comes from intuition.
Moreover, as far as I can see, that intuitive step contradicts basic facts, so according to the principles of science, it has to be discarded -- unless it can be shown that there might be something wrong with those basic facts. But it is up to you to prove this -- and with "proving" I mean the use of standard scientific procedures to maintain consistency.
My dear scholar
1.In mathematics "Intuitionism " is one accepted method of Philosophy.The other ones are Formalism, Logicism , Predicativism etc.
2. Pi and Pythagorean theorem are very basic and very ancient concepts.
3. So basic /fundamental concepts are ALWAYS simple , clear ,beautiful in their nature.
4. This author is a student of Zoology and taught the same in colleges till 2003..
His knowledge in mathematics is of High school level standard and that to only in Geometry ( 6th class to 12 th class : period 1956 - 1963 )
5. When he was a Lecturer he got a DOUBT in 1972 and questioned himself WHY NOT WE CALCULATE AREA & CIRCUMFERENCE OF A CIRCLE WITHOUT PI CONSTANT as in square and triangle .
6. He searched from 1972 to 1998. ( attachment)
7. He found the above 2 fundamental concepts are inseparable and are like two sides of a coin. (attachment )
8. As he is a BIG ZERO in his knowledge in mathematics every thing explained in his book PI OF THE CIRCLE ( 6 volumes ) enclosed - is new, original and unknown to any body and no where they are found.
It is difficult to work a petrol based automobile ENGINE with our Solar panel system. Similarly this work on pi is INCOMPATIBLE with the well established mathematical principles.
9. Hence this basic concept Pi is purely an intuitive in its basic methods.
10. Kindly understand my deficiency no no no complete IGNORANCE in mathematics .It is a truth and the mathematics world should read, study, dissect, refine this entire work and shape the whole concept on YOUR established PRINCIPLES of Mathematics. My DUTY is over. Your study BEGINS now.
OK , my dear scholar
rsj reddy ( 72)
Well, I know from my own work that sometimes intuition helps to find the next step. But intuition alone is never enough in science; all steps have to be supported by a valid reason -- because that's what science is about: to clarify the "why" at each step; only that way it all makes sense.
You suggest that "the mathematics world should read, study, dissect, refine this entire work" of yours -- and what I'm trying to do here is exactly that. However, it turns out that your work is missing a valid argument for a most relevant step. So, it doesn't follow the established principles of mathematics, and therefore, I guess, isn't of interest for the mathematics world.
Moreover, it seems that you overestimate the role of "truth" in science. Astonishingly, there is no "truth" in science in the sense of an absolutely true statement or an individually enlightening sentence; one can only say that a scientific statement hasn't been found wrong up to now.
You said that your "basic concept pi is purely an intuitive [one] in its basic methods" and that your "work on pi is incompatible with the well established mathematical principles." As such, it's not science. Therefore, the question that you started with -- "Why Does Circle-Square Composite Construction Rejects Present Pi Number 3.1415926...?" -- cannot be answered other than "No, it doesn't, because it's not scientific" -- unfortunately.
OK .then my dear scholar
Any way have a look at the following 2 attachments for what is wrong with the present pi
OK .then my dear scholar
Any way have a look at the following 2 attachments for what is wrong with the present pi
Yes, of course -- I've noticed these contributions already. Unfortunately, there is something wrong. Let me explain: As is discussed in "img027-1.jpg" (Calculus and Analytic Geometry), one constructs an approximation to an unknown value of a certain quantity by considering both an upper and a lower boundary for this quantity.
One does this in a way that allows to systematically approach the desired quantity step by step, and for each step the lower boundary becomes a little bit larger, while the upper boundary becomes smaller. Since this happens in each step, these boundaries may meet in the limit of infinite many steps; if they meet, both have approached the unkown value of the desired quantity. (If they don't meet, there is no well-defined limiting value, which means that this procedure doesn't help to determine the desired value.)
Now, compare this to what Vitthal Jadhav wrote in his e-mail ("img028.jpg"): He consideres a lower boundary only, because he writes: "[...] then the polygon look[s] like [a] circle, but we cannot deny that the sides of [the] polygon which is constructed by joining two adjacent vert[ices] [...] [are] smaller than the circular arc joining these two points [...]."
This is correct, but it it's not enough. He misses to consider the same for a circumscribing polygon (i.e. one for which the circle is perfectly inscribed). Also this polygon, for a very large number of vertices, looks much like the circle, but its sides are larger than the corresponding arcs. For an increasing number of sides, the difference to the circle becomes smaller and smaller.
It is clear that the unknown value of pi lies between the area values of these polygons -- the one that is inscribed in the circle (giving a lower boundary), and the one that circumscribes the circle (giving an upper boundary).
Now let's have a look at specific values of these polygons, namely for ones with 96 vertices. The total area for inscribed polygon is 3 + 10/71, and that of the circumscribed one is 3 + 10/70. This gives the condition that 3.140845... < pi < 3.142857... -- as already mentioned by Vitthal Jadhav, but unfortunately he missed the fact that the upper value is an upper boundary! The true value of pi lies below that boundary. Your cosmic pi value of 3,146..., however, lies above that upper boundary and therefore is too large.
On the other hand, in "img027-1.jpg" there is an arrow pointing to an equation that gives different boundary values for pi, 3.1466... being the upper one. However, this value holds only for the chosen step of the approximation -- which is, as is stated the text, the one for n = 10. So, what about this upper boundary value for larger n? Of course it will become smaller, the larger n gets; please check this yourself. In the end you will find a certain step with a large enough n so that also according to this error estimation your cosmic pi value will be too large.
If something is still unclear, or if you have further questions, don't hesitate to ask!
My Dear Scholar
1. Present Pi is polygon based
2. Polygon with its doubling of sides either inscribed or and circumscribed will NEVER become a circle
3. We are told parallel line meet at infinity. It is a wishful thinking based on our optical illusion.
4. Archimedes said UPPER range for pi is 22/7.
In his days there was no decimal system of numbering because he did not know 0 zero. He expressed in terms of a fraction 3+1/7.
1/8 is too small . 1/6 is too large.
5. It is called Exhaustion Method of Eudoxus of Cnidus of 400 BC. Archimedes refined his method.
6. We call this method is basrd on LIMIT concept.
7. Limit concept is not a proof
8.Limit concept is a LOGIC.
9. So, this method is NOT a proof to call that 3.141... as PI OF THE CIRCLE
10. No doubt 3.141...is a proof for polygon.
11. HOW CAN WE TAKE POLYGON's NUMBER AS PI OF THE CIRCLE?
IT IS WRONG 100% in Science.
12. Mathematics is NOT a science. It is an invented subject
13. Geometry is a science. For every concept there is a CONSTRUCTION
14.What is the definition of pi = It is circumference of circle/Diameter of circle
15.In limit concept what are we doing?
perimeter of polygon/Diameter of circle.
16. So, 3.141... VIOLATES the very definition of Pi
17.One more error no no no a BLUNDER
What is that blunder ?
Answer: the process of ROUNDING at every doubling of edges/ sides of the polygon
18. Square root of 3 we have to use.
19 We are taking just 10 decimals setting aside millions of decimals to the wind
20. Is it mathematics ?
21. In other words the EXHAUSTION METHOD OF ARCHIMEDES HAS ITS CREDIBILITY BECAUSE OF ROUND OFF OR TRUNCATION.
OK my dear scholar
My Dear Scholar
1. Present Pi is polygon based
2. Polygon with its doubling of sides either inscribed or and circumscribed will NEVER become a circle
3. We are told parallel lines meet at infinity. It is a wishful thinking based on our optical illusion.
4. Archimedes said UPPER range for pi is 22/7.
In his days there was no decimal system of numbering because he did not know 0 zero. He expressed in terms of a fraction 3+1/7.
1/8 is too small . 1/6 is too large.
5. It is called Exhaustion Method of Eudoxus of Cnidus of 400 BC. Archimedes refined his method.
6. We call this method is based on LIMIT concept.
7. Limit concept is not a proof
8.Limit concept is a LOGIC.
9. So, this method is NOT a proof to call that 3.141... as PI OF THE CIRCLE
10. No doubt 3.141...is a proof for polygon.
11. HOW CAN WE TAKE POLYGON's NUMBER AS PI OF THE CIRCLE?
IT IS WRONG 100% in Science.
12. Mathematics is NOT a science. It is an invented subject
13. Geometry is a science. For every concept there is a CONSTRUCTION
14.What is the definition of pi = It is circumference of circle/Diameter of circle
15.In limit concept what are we doing?
perimeter of polygon/Diameter of circle.
16. So, 3.141... VIOLATES the very definition of Pi
17.One more error no no no a BLUNDER
What is that blunder ?
Answer: the process of ROUNDING at every doubling of edges/ sides of the polygon
18. Square root of 3 we have to use.
19 We are taking just 10 decimals setting aside millions of decimals to the wind
20. Is it mathematics ?
21. In other words the EXHAUSTION METHOD OF ARCHIMEDES HAS NO CREDIBILITY BECAUSE OF ROUND OFF OR TRUNCATION.
OK my dear scholar
Ad 2.) "Polygon with its doubling of sides either inscribed or and circumscribed will NEVER become a circle" -- yes and no. Yes, because at any finite step with a certain number n of segments / vertices, both the inscribed and circumscribed geometrical objects are polygons, not a circle. No, because here we are talking about an infinite series of steps.
Therefore it doesn't matter at which of the various values of n you have a look at the geometrical objects under consideration, there are infinitely many refining steps to follow. Think about this! What does it mean for the deviation of the polygon from the circle? It means this: You may point at any position close to the circle, as close as you like, and of course there will by many n values for wich the polygon is at least that far away from the circle. However, this is always a finite number -- but since there are always infinitely many more refining steps, there will be infinitely many polygons which are closer to the circle than your chosen position. Since this holds for any position deviating from the circle, the only positions which are "save" in the sense that they never will lie "out of bounds" are those on the circle itself. Therefore, in the limit of infinitely many steps, we do reach the circle.
Ad 7.) "Limit concept is not a proof" -- in your eyes, but for mathematicians it is.
Ad 8.) "Limit concept is a LOGIC." -- This is in contradiciton to #7, because all proof is obtained by logic.
(I've sketched it above. In short, the argument is that all points which are not on the circle will at some point be either clearly isolated outside the outer polygon or clearly isolated within the inner polygon. So, logic tells us that the only remaining points are those on the circle itself.)
Ad 10.) "No doubt 3.141... is a proof for polygon." -- Not at all! On the contrary, no doubt that the polygon proofs cosmic pi (which equals 3.146...) wrong! To see this, here is a nice small thought experiment for you: Take a rope around the earth's equator (in the thought experiment taken to be a circle) such that it lies flat on the ground, then add to its length one full meter. About how many centimeters do you have to lift it from the ground everywhere so that it forms a circle again? (For this calculation it doesn't matter which pi value you use.)
My Dear Scholar
Our Cosmos is made of infinite number of quarks. Do you mean to say a quark at infinity becomes some thing else?
I am SORRY I am a total failure in arguments
Have a strong belief in your arguments. OK
Cosmic Pi is NOT my number
It was there before my birth
Whether you or world of mathematics accepts it or rejects it I do NOT care.
Truth can be suppressed citing arguments. But it can not be destroyed.
TRUNCATION in Archimedes method is simply ignored.
OK my dear scholar.
"Have a strong belief in your arguments. OK" -- No, that's not what science is about! Arguments stand for themselves; anybody can check them, and maybe some day they are disproved. (This is one aspect of what you expressed as "Science is a self-correcting one".)
"Cosmic Pi is NOT my number" -- Sorry, I never meant it personally! I just wanted to say that a value of 3.1464466... is too large; it cannot be pi since it is disproven by a circumventing polygon. Therefore, there is and never was such a thing as "cosmic pi = (14 - sqrt{2}) / 4". This is so because this value makes no sense -- and here is why, strictly geometrically:
Build a rather large circle, having a diameter of 10 km. This should be possible within a tolerance of 1 m, so the accuracy would be 1 m / 10,000 m = 0.01 %. Then, the circumference will have a length of pi times 10 km (within this tolerance), i.e. it will be approximately 31 km long. (For comparison, the tunnel of the Large Hadron Collider has a length of about 27 km, so in general, such large circles can be built.)
Then, measure the circumference of this circle within the same tolerance of 0.01 %, which corresponds to about 3 m. This would enable you to discriminate between a length of 31,416 m and 31,464 m, because the difference between these two amounts to 48 m, which is well above the tolerance limit. You would find that the circle with a diameter of 10,000 m has a circumference of 31,416 m. The reason is that 31,416 m are enough to form a polygon that circumferences the given circle, touching the circle only tangentially.
Or, consider it differently: Build two large circles with prescribed circumference legths, one with 31,416 m and one with 31,464 m, and measure the resulting diameter. One of these circles will have a diameter of 10 km (exact up to 1 m), the diameter of the other will be larger by approximately 14 m -- for the same reason as mentioned above (circumferencing polygon).
"TRUNCATION in Archimedes method is simply ignored." -- No, not at all! This is automatically taken care of in the polygon approximation limit. And polygon approximation, including truncation, is sufficient to prove "cosmic pi" wrong.
Dear Scholar
The Circumscribed polygon about a circle is not a PERFECT ONE in the sense , its edge/ side does not stay outside the circumference but it INTRUDES into the circle. It would look like a golden ring on the finger . At the time of wearing ring the finger might be just sufficient. But after some time that individual would have added some fat . But still that ring remains intruding the finger.
I always say each side of the circumscribed polygon about a circle is a tangent which should JUST touch the circumference.
But in reality this tangent CROSSES the circle and crushes the circle to some extent.
This is the reason why its perimeter is also 3.141.. similar to the perimeter of the inscribed polygon in the circle. In 36-gon the perimeter of the circumscribed polygon about a circle is 3.146...
In the next doubling it becomes erratic.
To conclude the value 3.141... is WRONG
Dear scholar
You are WRONG in saying all the million decimals of root 3 are included in calculations. Even Super computers can not take so many decimals.
TRUNCATION is ignored and hence 3.141... of both inscribed and circumscribed in and about circle is a LOW VALUE. The Real value is 3.146...
"But in reality this tangent CROSSES the circle and crushes the circle to some extent." -- You're funny! Look at the drawings in your own documents (that you have posted here, containing your "derivation" of "cosmic pi"), showing a circumscribed square. Do you see any crossing or crushing there? I don't!
"In 36-gon the perimeter of the circumscribed polygon about a circle is 3.146..." -- Well, I haven't checked that yet; do you have the relevant calculation at hand and could post it here, too? Or is it included in the material that you've already posted here?
"In the next doubling it becomes erratic." -- Why? What makes the next doubling so fundamentally different from the 36-gon one, geometrically?
"You are WRONG in saying all the million decimals of root 3 are included in calculations." -- Who said that? I didn't! Maybe this answer belongs elsewhere?
"TRUNCATION is ignored and hence 3.141... of both inscribed and circumscribed in and about circle is a LOW VALUE." -- No: Since truncation isn't ignored, there is nothing wrong with the value 3.141... of pi.
"The Real value is 3.146..." -- No, this value is wrong, as proven by the circumscribed polygon. See the work of Archimedes: 22/7 = 3.1428... is an upper limit for pi.
Dear Scholar
TRUNCATION is a fact. Millions of decimals of root3 are omitted in calculations . It is Calculation flaw. So, Exhaustion method is a TRUNCATION method. Hence this lower value 3.141...instead of 3.146.....
The Circumscribed Method is UNSUITABLE in deciding the value of pi.
HERE IS MY CHALLENGE TO YOU & TO THE WORLD OF MATHEMATICS
I AGREE if you obtain 3.141.... from the RADIUS of circle alone.
Can you disprove the enclosed method?
OK my dear
Stop: You have started to claim that, against all odds, the value of pi should be 3.146... rather than 3.141... -- so it's up to you to prove this. However, all what you've presented so far was invalid; several people have judged this independently.
So, the questions go to you again:
"The Circumscribed Method is UNSUITABLE in deciding the value of pi." -- Why? What makes the circumscription so fundamentally different from the inscription, geometrically?
"In 36-gon the perimeter of the circumscribed polygon about a circle is 3.146... In the next doubling it becomes erratic." -- Why? What makes the next doubling so fundamentally different from the 36-gon one, geometrically?
Dear Scholar
In the inscribed polygon its side is a CHORD of the circle.. It remains inside the circle.
In the circumscribed polygon its side is a TANGENT of the circle. It DOES NOT TOUCH the circumference ONLY at its MID-POINT . The tangent does NOT remain OUT SIDE of the circle. The side of polygon ( tangent ) MERGES with the circumference.
No doubt its perimeter IS 3.141...
ALL the tangents MERGES with the circle and hence its value 3.141...does not APPLY to, as perimeter of the circle.
This is the reason for its UNSUITABILITY in computing PI OF THE CIRCLE.
You may see the diagram in any book on Pi
OK
"In the circumscribed polygon its side is a TANGENT of the circle. It DOES NOT TOUCH the circumference ONLY at its MID-POINT." -- Wrong!! It is a constitutive property of a tangent to just have one point in common with the circle.
"The tangent does NOT remain OUT SIDE of the circle." -- Wrong!! If some straight line does not reamin outside of the circle, it has two intersection points with the circle, and then it can't be a tangent anymore.
"The side of polygon (tangent) MERGES with the circumference." -- Impossible: The circle is curved, and the tangent is a straight line, so there is no merging possible. (If there were some merging in a certain interval, the circle wouldn't be curved in that interval, so it wouldn't be a circle.)
Conclusion: You got the idea of a "tangent" completely wrong.
Dear Scholar
1. For 1st you are mistaken. You may look at the diagram
2.For 2nd : That is what I am saying
3.For 3rd: You may see the diagram with your own eyes.
If possible I will send the diagram from a book on Pi
OK
Dear Sarva Jagannadha Reddy, thanks a lot -- these answers of yours are an eye-opener for me! (For all others: This is about a diagram which is related to Cusanus' method to approximate pi; it was sent directly to me from S. J. Reddy, therefore it is not visible publicly.)
(Edit note: Meanwhile S. J. Reddy has posted this diagram in the Comments to the following item: https://www.researchgate.net/project/Discovery-of-the-TRUE-VALUE-of-Geometrical-Constant-COSMIC-Pi?_updateId=5996b6d7b53d2ff30bda8c8e&replyToId=5a55825c4cde266d58818b31)
(Further edit: Here's a direct link to the diagram: https://www.researchgate.net/messages/attachment/1216799_img129.jpg)
Ad 1.) Be aware that there are several circles in this diagram. The polygon side XY is a tangent only for the second-innermost circle, passing through point Q and named "c2n in". As such, it remains outside this circle.
Ad 2.) The same polygon side XY is an intersecting line for the second-outermost circle, passing through the points X and Y and named "c2n circum". As such, it's not a tangent for this circle.
Ad 3.) There is no real merging of the tangent with the circumference; this is just a visual artefact coming from the different line thicknesses. Imagine the tangent being drawn as thin as the circle, then it becomes clear that the only overlap between them is at point Q.
Beware, my dear Sarva Jagannadha Reddy, that in this diagram, there are circles of different diameter. So, one has to be careful when talking about the successive approximation of a circle by inscribed and circumscribed polygons -- because in my mind there is only one circle, having a fixed diameter. But it seems that in your mind, you referred to this diagram; maybe this is why you stated that "In 36-gon the perimeter of the circumscribed polygon about a circle is 3.146... In the next doubling it becomes erratic"?
Dear Scholar
I can not accept 3.141.. of polygon as pi of the circle.
Why?
I can not claim some body's shirt as mine with arguments.
"I can not accept 3.141.. of polygon as pi of the circle" -- Now it's you who is reluctant!
"I can not claim some body's shirt as mine with arguments." -- This is a confession, which means that it belongs to religion, but not to science.
"I can not claim some body's shirt as mine with arguments." -- Well, let me say it again: This is not about any specific person's arguments, but just about mathematical logic -- which is independent of the person using it.
However, of course there is one personal aspect in mathematics: If somebody claims something new, it's up to him to prove the correctness of his claim. So, the remaining question goes to you again:
"In 36-gon the perimeter of the circumscribed polygon about a circle is 3.146... In the next doubling it becomes erratic." -- Why? What makes the next doubling so fundamentally different from the 36-gon one, geometrically?
OK? So also here we end up in the following state (as you wrote in the project comments; see the link given below):
"Let it it be the Cosmic Pi is WRONG
I told you yesterday ' I do not care ' "
(https://www.researchgate.net/project/Discovery-of-the-TRUE-VALUE-of-Geometrical-Constant-COSMIC-Pi/update/5996b6d7b53d2ff30bda8c8e?replyToId=5a54b8824cde266d588154a8&_iepl%5BviewId%5D=RDFW2EdNTdp6f7B6tYLtjYZe&_iepl%5BsingleItemViewId%5D=kr2tgqgt4Q07dBicAX1Ix69C&_iepl%5BinteractionType%5D=projectUpdateView)
"In 36-gon the perimeter of the circumscribed polygon about a circle is 3.146... In the next doubling it becomes erratic." -- Why? What makes the next doubling so fundamentally different from the 36-gon one, geometrically?
Dear Scholar
Light travels straight from the Stars till that light enters the atmosphere of the Earth. For that reason we sing " twinkle twinkle little star..."
"After seeing 5-gon diagram sent by you I am convinced even at this stage the Circumscribed Polygon does NOT NOT NOT remain out side the circle completely. For this I am very thankful to you." -- Well, so what about the square ABCD that you have drawn in your methods: Does it also not completely remain outside the circle?
Dear Scholar
No, square does not remain out side the circle.
Because , the side of the square and the diameter of the circle is SAME.
For that reason I used to say in the past circle is not inscribed IN the square.Then diameter is less than side.
But, I was saying in the past " circle is inscribed WITH the square.
Diameter is equal to side.
-------------------------------------------------------------------------------------------------------------
If circle is inside the square my entire work collapses like a structure of playing cards .
SIDE= DIAMETER is the soul of my ENTIRE work.
It means the term tangent for side is NOT apt.
I thought for many years on this aspect and found so many different definitions for the term tangent and confusing.
I ignored it NOT to confuse the reader
____________________________________________________________
By the by, to be frank whole work done till March 1998 was all an effort to make wrong as right.
Dear Sarva Jagannadha Reddy, thanks a lot for these explanations; they are very wise, I like them a lot -- especially that you prefer to say that a "circle is inscribed WITH the square", because then it's clear that the circle's diameter is equal to the square's side. And it is wise to avoid things which might confuse the reader.
"SIDE = DIAMETER is the soul of my ENTIRE work." -- Yes, I see! Moreover, from that it is also clear that such a square does not remain outside the circle, but it touches the circle exactly where the diameter corresponds to a side (as, e.g., in the 145th method where JK and LN demonstrate this fact). And it is clear that the square touches the circle only there, because at any other point on the circle, the diameter is tilted and does not correspond to a side. Do you agree?
Now, let's consider two identical squares (i.e., their side length is the same), having circles inscribed with the squares. So, these two circles are also identical (i.e., their diameter is the same).
Both squares are 100% perfect for the inscribed circles, and each square and circle are aligned perfectly and naturally. And since a circle is round and perfectly symmetric, this holds for any orientation of the square with respect to the cicle; it doesn't matter whether the square is oriented vertically, or whether it is tilted.
Now, let's consider one of the squares being oriented vertically, the other being tilted by 45°. Both inscribed circles are identical, so these two can be used to join the two circle&square combinations by identically overlapping the two circles. Thereby we would end up with a circle that is circumscribed by two squares, which are tilted with respect to each other by 45°, and the circle would be perfectly inscribed with both of them simultaneously. Do you agree?
This is a natural construction, because both squares are 100% perfect for the inscribed circles, and each square and circle are aligned perfectly and naturally. So, after overlapping the circle would be perfectly inscribed with both squares simultaneously. Do you agree?
You can easily produce it yourself: Take two sheets of paper and make the drawing of a circle being inscribed with a square (as you did in all your methods) on both of them, in exactly the same size. When finished, put one sheet behind the other and hold them together against the light. Then, adjust the sheets to make the drawings overlap exactly. Finally, turn one of the sheets by 45° so that the circles still overlap exactly, but the squares become visible separately.
Of course, it is favorable to use transparent paper for this construction right from the beginning; this also makes it easier to increase the number of layers later on.
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