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Circle Squared?


Robust

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I would say so, yet find my own colleagues so adamantly opposed to even the suggestion of such possibility as refusing even to discuss the matter. So....off to the forum!

 

I do have a set of formulae describing a squaring of the circle; yet, not being a maths person myself,per say, would appreciate any critique I can get before presenting a paper on the subject. Might there be the interest here?

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Given first consideration was the assumption that whatever the configuration it must relate to pi in all regards. Thus, first given as relating pi to the square is the pi quadrature formula which states:

 

1) sqrt(pi)*r gives the side of a square with the same area as that of the given circle. I give as a standard in my configuring a diameter of 9 units (for reasons as will be shown), thus a superinscribed square with area of 63.617. One might think that would be enough, yet the protagonists say no - and do present reasonable arguement - and so we must push on. I would suggest that those wishing to follow the theme draw the superinscribed circle to scale on a sheet of graph paper.

 

2) area/[pi/2] = area of inscribed square, the angles of which define the chord length to each quadrant the circle. Chord length is important to the final solving.

 

3) Trianglature Formulae: a) sqrt(2)*r = quadrant chord; :) quad chord*pi/4*sqrt(2) = length of chord on the arc. Now we turn to the length of that line of the superinscribed

square that intersects the arc and as extending on either side of the arc. What I see in this is the scriptural reference to "The 4 corners of the Earth." What I believe da Vinci saw and related to in his famous drawing was a relationship of the phi-pi ratios as pertaining perhaps to the spiral. For I suspect that if one moved the square of his drawing up to where its lines intersected the arc equally we would find the superinscribed circle.

 

Just a start and with a number of interesting formulae remaining. I would like to and will be happy to see it through here if you wish.

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Originally posted by: Robust

Sorry about the format with lack of paragraphs. Not the way it was written and don't know how to remedy. Help, please! </P>

 

 

Some people seem to have this issue. What platform/browser are you on?

 

Try to insert P tags, it might help (as in < P > without the spaces).

 

If you still have problems, let me know.

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1) sqrt(pi)*r gives the side of a square with the same area as that of the given circle.

ok true

 

give as a standard in my configuring a diameter of 9 units

I wouldn't do that; Dont use 'real' numbers, unless your completely finsihed (it obscures in a way what you are doing).

 

 

2) area/[pi/2] = area of inscribed square, the angles of which define the chord length to each quadrant the circle.

 

so area/(pi/2)=2r^2... i dont really understand what the 'inscribed square' is... And i als dont understand how an area is related to an angle...

hmm i actually think i cant follow much of the rest of your story maybe if you can clarify it with a drawing?

 

What I believe da Vinci saw and related to in his famous drawing was a relationship of the phi-pi ratios as pertaining perhaps to the spiral.

 

As far as i know this has to do with the 'golden cut'

 

Bo

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Thanks much Tormod. I'll continue on then....

 

Here is the most interesting formula of them all. It derives from an intensive archeological study into origin of the Base 10 number system 0-9 and from which system it might be speculated on that Pythagoras derived his system of perfect ratios - where the cipher represents Base 10.

 

4) D^2 (10) (DD) = area; where D is diameter; (10) is Base 10; DD the degree-distance between each adjacent angular degree on the circumference as given by the cardinal number 9; i.e., 0.0785....ad infinitum.

 

Example: 9^2 (10) (0.0785....) = 63.617....area;

8^2 (10) (0.0785....) = 50.265....area;

7^2 (10) (0.0785....) = 38.484....area;

 

And so on for any diameter whatever.

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Bo, Hello!

 

The inscribed square is simply the chord lengths of thecircle's 4 quadrants. And I'd truly like to give a drawing, but I'm not that computer literate. The area of the inscribed square is simply Area divided by 1/2 pi.Its corners descrine the circle's 4 cardinal points.

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I think Robust wants to prove that you can square the circle. The inscribed square is just one step on the way.

 

However, it has been proved that it is impossible. Here is one explanation:

 

Squaring the circle

http://www.geom.uiuc.edu/docs/forum/square_circle/

 

So, Robust, in order to prove your theory you also need to prove why the proof on that page is wrong.

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Thanks for the drawing, Bo...very nice!

 

What it appears you have there is the superinscribed sqquare minus its extensions beyond the arc. What formula did yo use in drawing it?

 

The inscribed square I gave, however, defines the cardinal points given of the quadrants by the formula sqrt2*r which gives the quadrant chord length.

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This is the 2nd time now already you have come to my aid, Tormod....it appears I have come to the right place.

 

Thanks much for the web site. I'll go there directly. I do realize that this is a highly contentious arguement I am making, so looking for the best opposition I can find.

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Originally posted by: Robust

Tormod,

 

I find the Crowder/Robbins study you recommend to be one of the better. The equations will have to be given more time on my part, yet I do have a major contention with its summation. Will get back to y'all on it.

 

THere's others of us out here that would enjoy your insights as you battle this out.

 

Thanks

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Good to know, Freethinker....and, yes, I do suspect it will be a battle of sorts.

 

The Euclidean geometry I read of the Crowder/Robbins study will not do, because we know with good certainty that there is no such thing as the absolute straight line; consequently, the notion of a perfect circle is brought to question also - all of it seeming to bear upon the fact that neither have we yet discerned the absolute static condition.

 

It is my opinion that the bestthat might be hope for is to determine a ratio between the line and the arc, which we have succeeded in doing by the pi ratio. A squaring of the circle then might seem to depend upon the consistency of all line and arc segments of the circle's superinscribed square conforming in every instance to the one ratio.

 

Most importantly (as I see it) I wish to know of any criticism of the given Base 10 formulae. I suspect my entire approach hinges on its veracity.

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I see it as no coincidence that the degree-distance given of a diameter of 9 is pi/40. I likewise have to believe it was on that realization by Pythagoras, which he no doubt learned from the Babylonians, that he developed his system of perfect ratios (0:1:1:2:3:4) where the cipher represents Base 10 and 4 the culminating ratio to all possibilities for describing an exacting mathematical relationship. I have to believe that together with the pi ratio, the anomaly shown by the Base 10 formulae says it all; and likewise, being as how every dimensional relationship of line to arc of the circle's superinscribed square correlate by pi....the circle IMHO is squared.

 

It's truly worth the consideration. Our International Space Agencies are in the planning stages of sending a probe beyond our solar system, the results of which we will not know in our lifetime and probably many lifetimes to come. It is not my opinion alone suggesting that except we are able to define the coordinates to some 7 decimal places....there will be no destination.

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