DEUS = PI (π)


ARITHMETIC DEMONSTRATION FROM THE CONCEPT OF GOD:
The Ratio From The Perfection
The Number PI
(π)

Archimedes obtained the first rigorous approximation of $\pi$ 
by Circumscribing and Inscribing $n=6\cdot 2^k$-gons on a Circle.

$\displaystyle a(n)$$\textstyle =$$\displaystyle 2n\tan\left({\pi\over n}\right)$
$\displaystyle b(n)$$\textstyle =$$\displaystyle 2n\sin\left({\pi\over n}\right),$

Successive application of Archimedes’ Recurrence Formula 
gives the Archimedes algorithm,

which can be used to provide successive approximations to $\pi$ (Pi)

The first iteration of Archimedes’ Recurrence Formula then gives

$\displaystyle a_1$$\textstyle =$$\displaystyle {2\cdot 6\cdot 4\sqrt{3}\over 6+4\sqrt{3}} = {24\sqrt{3}\over 3+2\sqrt{3}} = 24(2-\sqrt{3})$
$\displaystyle b_1$$\textstyle =$$\displaystyle \sqrt{24(2-\sqrt{3}\,)\cdot 6}=12\sqrt{2-\sqrt{3}}$
 $\textstyle =$$\displaystyle 6(\sqrt{6}-\sqrt{2}\,).$

Additional iterations do not have simple closed forms,
but the numerical approximations for $k=0$, 1, 2, 3, 4
(corresponding to 6-, 12-, 24-, 48- gons) are

<img width="157" height="25" src="https://archive.lib.msu.edu/crcmath/math/math/a/a_1371.gif&quot; alt="\begin{displaymath} 3.00000 < \pi
<img width="157" height="25" src="https://archive.lib.msu.edu/crcmath/math/math/a/a_1372.gif&quot; alt="\begin{displaymath} 3.10583 < \pi
<img width="157" height="25" src="https://archive.lib.msu.edu/crcmath/math/math/a/a_1373.gif&quot; alt="\begin{displaymath} 3.13263 < \pi
<img width="157" height="25" src="https://archive.lib.msu.edu/crcmath/math/math/a/a_1374.gif&quot; alt="\begin{displaymath} 3.13935 < \pi