Π day

[Thomas Klemm]

Just stumbled upon this visual proof of Gregory’s Theorem:

Quote:Let \(I_k\) and \(C_k\) denote the areas of regular \(k\)-gons inscribed in and circumscribed around a given circle.
Then for all \(n\), \(I_{2n}\) is the geometric mean of \(I_n\) and \(C_n\), and \(C_2n\) is the harmonic mean of \(I_2n\) and \(C_n\); that is,

\(
\begin{align}
I_{2n}=\sqrt{I_{n}C_{n}}
\end{align}
\)

and

\(
\begin{align}
C_{2n}=\frac{2}{\frac{1}{I_{2n}}+\frac{1}{C_{n}}}.
\end{align}
\)

So I thought I give it a try using the || operator of the WP-34S:

Code:

001 RCL× Y
002 √
003 ||
004 STO+ x
005 RCL L
006 END

Initialize the stack:

4
ENTER
2
RTN

Calculate the sequence:

R/S
R/S
R/S
R/S
…

We end up with the same sequence as when using Vieta's formula.
Hint: Select YDON in MODE to display both \(I_k\) and \(C_k\).

3.3137085
 2.82842712475

3.18259788
 3.06146745892

3.15172491
 3.12144515226

3.14411839
 3.13654849055

3.14222363
 3.14033115695

3.14175037
 3.14127725093

3.14163208
 3.14151380114

3.14160251
 3.14157294037

3.14159512
 3.14158772528

3.14159327
 3.14159142151

3.14159281
 3.14159234557

3.14159269
 3.14159257658

3.14159266
 3.14159263434

3.14159266
 3.14159264878

3.14159265
 3.14159265239

3.14159265
 3.14159265329

3.14159265
 3.14159265351

3.14159265
 3.14159265357

3.14159265
 3.14159265359

3.14159265
 3.14159265359