Quadrature of a circle solved

Remember that there are also 2 other modern values of Pi in addition to the traditional most common value of Pi that is 3.141592653589793. The 2 other values of Pi are: 3.146446609406726 and also 3.14460551102969.
Nonsense. The second two numbers are not $$\pi$$.

3.146446609406726 is the result of 14 subtract square root of 2 = 1.414213562373095 = 12.585786437626905 and then 12.585786437626905 is divided by 4.
Why do you think that
$$\pi = \frac{14-\sqrt{2}}{4}$$?
How did you derive that result? (Just to emphasise: it is incorrect.)

3.14460551102969 is the result of 4 divided by the square root of the Golden ratio = 1.272019649514069.
Why do you think that
$$\pi=\frac{4}{(1+\sqrt{5})/2}$$?
How did you derive that result? (Just to emphasise: it is incorrect.)

Please remember that the Golden ratio can be obtained in Trigonometry through the formula Cosine (36) multiplied by 2.
What is the relevance of this?

All modern Pi values must be approximated meaning the results that are given to us by the 3 modern Pi values must be reduced to 4 or 5 decimal places resulting in multiple approximations for the 3 main Pi values.
What?!

There's only one "modern pi value". Why 4 or 5 decimal places?

There is a debate among scholars regarding which version of the 3 values of Pi are better to use when dealing with circles and geometrical figures.
No. There is no debate. Only one of the values you have cited approximates pi.
 
Why do you think that
$$\pi=\frac{4}{(1+\sqrt{5})/2}$$?
Actually, it was $$ \frac{4}{\sqrt{\frac{1 + \sqrt{5}}{2}}} = 4 \sqrt{\frac{2}{1 + \sqrt{5}}} = 2 \sqrt{ 2 ( \sqrt{5} - 1 ) } $$

It is almost impossible to create a circle and a square with equal areas of measure with 100% accuracy due to the transcendental nature of the modern traditional version of the ratio Pi 3.141592653589793.
Not "almost" -- 100% accuracy is never found in any manufacturing technique. It is also impossible in Euclidean geometry to construct circles and squares of equal area. Likewise it is impossible to form an exact expression in terms of just the four arithmetic operators and roots or to find a polynomial equation with integer coefficients that gives pi as a root.

But the approximate forms you give are not very impressive as intellectual achievements.

$$\frac{14 - \sqrt{2}}{4}$$ gets the first 3 digits right. If a circle had a diameter of 12 million meters (about the diameter of the Earth) the error in the circumference calculated with this approximation would be about 58,247 meters too long.
$$ \frac{4}{\sqrt{\frac{1 + \sqrt{5}}{2}}}$$ gets the first 3 digits right, but is better than the above.
3.14 gets the first 3 digits right, but is better than the above. If a circle had a diameter of 12 million meters (about the diameter of the Earth) the error in the circumference calculated with this approximation would be about 19,112 meters too short.
22/7 gets the first 3 digits right, but is better than the above. If a circle had a diameter of 12 million meters (about the diameter of the Earth) the error in the circumference calculated with this approximation would be about 15,178 meters too long.
$$\sqrt{ \frac{40}{3} - \sqrt{12}}$$ (Kochanski's Approximation) gets the first 5 digits right.
355/113 gets the first 7 digits right. If a circle had a diameter of 12 million meters (about the diameter of the Earth) the error in the circumference calculated with this approximation would be about 3.2 meters too long. That's likely as good as one would ever need for civil engineering.
$$\frac{99^2}{2206 \sqrt{2}}$$ (Ramanujan) gets the first 7 digits right.
$$\frac{66 \sqrt{2}}{ 33 \sqrt{29} - 148 } $$ (Borwein and Bailey) gets the first 8 digits right.
3.141592653589793 is a simple truncation of first 16 digits of the infinite decimal sequence for pi.
Due to design considerations, most PC and Mac programs are coded with IEEE double precision numbers which forces them to best approximate pi as $$\frac{884279719003555}{2^{48} }$$ which is a bit better than the above decimal truncation. This makes me suspect you cribbed the above from the computer without any heavy lifting on your part.
21053343141/6701487259 gets the first 21 digits right.
$$ \frac{ \ln ( (640320^3 + 744)^2 - 393768 ) }{ 2 \sqrt{163}} $$ (Warda) gets the first 46 digits right. But this is using the natural logarithm which is outside of the rules above.
But if we are allowing that, then we could use this: $$\frac{ \ln ( -1 ) }{ \sqrt{-1} } = \pi$$ and that is exact.
 
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