Welcome to Aquarius, Volume 7 (August 10, 2006)

Six Nines

Constructing Pi to the level of precision of 0.99999979...

__The simple fact of asking the question__:

What is most fascinating about constructions such as these, in pursuit of "squaring-the-circle," is that the differences between two line lengths are not visible to the naked eye. This search for a solution reveals the simple fact that the original riddle cannot be answered, no matter how much work is done, unless one knows the precise value of Pi. Therefore, this simple fact raises the question as to how and why the ancients would have posed the riddle in the first place, because to ask "Can we square the circle?" is to ask, necessarily, "Do we know the value of Pi?" Without the true value of Pi, there can be no true solution. That the question would be asked, implies that the asker has the value of Pi. This is like someone asking which is the highest of three mountains. The question implies that the asker knows how to measure the height of a mountain.

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Citation suggested: Manimas, John. Six Nines, The Rediscovery of Proportion, "Welcome to Aquarius," www.jmanimas.com, Volume 7, August 10, 2006.

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The construction steps for constructing Pi to the level of precision of 0.99999979...

__Part 1: Construction of line A__ = 0.392836323023349590666713745392125... [ * X] a value that is very near in value to Pi/8.

__Step 1__: Construct the 36-54 degree right triangle. If you need more information about this right triangle, it is described here in *Precision of the Ancients*, and labeled as Right Triangle #5. Divide the 36-degree angle in half. Let that new dividing line, which created an 18-degree angle, be the new hypotenuse for our first right triangle. Mark off a length of 1 on the new hypotenuse. From that upper vertex on the hypotenuse, construct a perpendicular down to the new base. You now have an 18-72 degree right triangle.

Initial construction drawing to produce "line A":

You now have a right triangle with the following side lengths and ratios:

A= 0.309016994374947424102293417182819...

B= 0.951056516295153572116439333379382...

H= 1.000...

Sine= 0.309016994374947424102293417182819...

Cosecant= 3.23606797749978969640917366873128...

Cosine= 0.951056516295153572116439333379382...

Secant= 1.05146222423826721205133816969575...

Tangent= 0.324919696232906326155871412215134...

Cotangent= 3.07768353717525340257029057603691...

We see that the cosine, B/H, equals 0.951056516295153572116439333379382...

and the secant, H/B= 1.05146222423826721205133816969575...

__Step 2__: Our next construction goal is to square the secant value and then double it.

__Step 3__: Re-construct this right triangle by constructing the horizontal base B and then the 18-degree angle. Mark off the length of 1 on the horizontal side B, and then raise the perpendicular to construct side A. Because side B has a length of 1, and we know that the cosine=0.951056516..., the length of the hypotenuse must now be 1.051462224... .

__Step 4__: Re-construct the 18-degree right triangle beginning as in previous construction, but mark off the length of 1.051462224... on the horizontal base B, and then raise the perpendicular to construct side A. The secant is again 1.051462224... and the length of the hypotenuse is now 1.051462224... * 1.051462224... or 1.10557280900008412143633053250749... , as follows:

cosine = __B__ = __1.05146222423826721205133816969575...__

H = 1.10557280900008412143633053250749...

= 0.951056516295153572116439333379382...

secant = __H__ = __1.10557280900008412143633053250749...__

B = 1.05146222423826721205133816969575...

= 1.05146222423826721205133816969575

__Step 5__: Next construct a sufficiently long straight line and mark off the length of our H twice, now our new line = 2.21114561800016824287266106501498...

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__Step 6__: The next step is to construct a line length of the square root of 8, which is 2 times the square root of 2, and which is from our previous constructions described in *Precision of the Ancients* (POA).

sqrt(8) = 2.8284271247461900976033774484194...

__Step 7__: We now subtract the shorter line 2.211145618... from 2.828427124... and the result is 0.617281506746021854730716383404417... . Our next construction goal is to construct the square root of this line length of 0.617281506... . The resulting line length will then be very near to Pi/4: 0.78567264604669918133342749078425... .

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__Step 8__: Construct a rectangle with a horizontal length of 2 and a width of 0.617281506... . Square this rectangle using the standard squaring-the-rectangle procedure, which is described in POA and in many geometry books. The area of this rectangle and constructed square is 2*0.617281506..., and the side of the constructed square therefore has a length that is equal to [sqrt(2) * sqrt(0.617281506...)] or [(1.414213562...) * (0.785672646...)].

__Step 9__: Is already completed because we want a square with the side length constructed in our previous Step 8. Our square with a side length of [(1.414213562...) * (0.785672646...)] has a diagonal with length 2 * 0.78567264604669918133342749078425... and this is equal to length 1.5713452920933983626668549815685..., very near to Pi/2. We then double the length of our diagonal for the final length of line A: 3.142690584186796725333709963137... . This length for line A is Pi * 1.00034948216336988124925683259591..., as follows:

sqrt(0.617281506...) = 0.78567264604669918133342749078425...

* 4 = 3.142690584186796725333709963137...

= Pi * 1.00034948216336988124925683259591...

__Part 2: Construction of average of (line A plus line B), where line B is our previously constructed line length (described in POA) of 3.14155655389944902222112102645257...__ .__ __

__Step 1__: Construct a sufficiently long line and mark off the length of our line A: 3.142690584186796725333709963137... . Then mark off the length of line B:

+ 3.14155655389944902222112102645257... (line B) and divide in half. Repeat 5 times:

-----> = 6.28424713808624574755483098958957... /2 =

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|<-----------new line ------------------->| (Step 1)

__Step 1__: 3.14212356904312287377741549479479...

+ 3.14155655389944902222112102645257...

-----> = 6.28368012294257189599853652124736... /2 =

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|<-----------new line ------------------->| (Step 2)

__Step 2__: 3.14184006147128594799926826062368...

+ 3.14155655389944902222112102645257...

-----> = 6.28339661537073497022038928707625... /2 =

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|<-----------new line ------------------->| (Step 3)

__Step 3__: 3.14169830768536748511019464353813...

+ 3.14155655389944902222112102645257...

-----> = 6.2832548615848165073313156699907... /2 =

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|<-----------new line ------------------->| (Step 4)

__Step 4__: 3.14162743079240825366565783499535...

+ 3.14155655389944902222112102645257...

-----> = 6.28318398469185727588677886144792... /2 =

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|<-----------new line ------------------->| (Step 5)

__Step 5__: 3.14159199234592863794338943072396...

and 3.14159199234592863794338943072396... / Pi = divided by

---> 3.1415926535897932384626433832795...

= 0.999999789519540719278826808398938...

inverse: 1.00000021048050358275423692719123...

Difference: 0.0000006.6124386460051925395255554217116...

Precision to: 3.1415926535897932384626433832795

__-3.14159199234592863794338943072396 __

= 0.000000661 ten millionths of a unit

__Part 3__: Referring again to *Precision of the Ancients*, under the __Pi and One Set of right triangles__, and to *The Myth of Exactly Pi*, we see that the ratio of our square area to our circle area will be the same as the ratio of our working value for Pi to our infinite value for Pi, and our ratio now is 0.999999789519540719278826808398938... . Does one's ability to get closer with persistence mean that squaring the circle *exactly* is possible?

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