Metodo di Esaustione (Quadrato)

Osserva

Numero lati	4	8	16	…
$l$ , lato	$\displaystyle \sqrt{2}\cdot r$	$\displaystyle \sqrt{\left(\frac{l_4}{2}\right)^2+\Delta_4^2}$	$\displaystyle \sqrt{\left(\frac{l_8}{2}\right)^2+\Delta_8^2}$	…
$p$ , perimetro	$\displaystyle 4\cdot l_4$	$\displaystyle 8\cdot l_8$	$\displaystyle 16\cdot l_{16}$	…
$\pi$ , pi greco	$\displaystyle \frac{p_4}{2 \cdot r}$	$\displaystyle \frac{p_8}{2 \cdot r}$	$\displaystyle \frac{p_{16}}{2 \cdot r}$	…
$a$ , apotema	$\displaystyle \sqrt{r^2-\left(\frac{l_4}{2}\right)^2}$	$\displaystyle \sqrt{r^2-\left(\frac{l_8}{2}\right)^2}$	$\displaystyle \sqrt{r^2-\left(\frac{l_{16}}{2}\right)^2}$	…
$\Delta$ , delta	$\displaystyle r-a_4$	$\displaystyle r-a_8$	$\displaystyle r-a_{16}$	…

Sostituendo il valore $\displaystyle l_4= \sqrt{2}\cdot r$

Numero lati	4	8	16	…
Lato	$\displaystyle \sqrt{2}\cdot r$	$\displaystyle \sqrt{2-\sqrt{2}}\cdot r$	$\displaystyle \sqrt{2-\sqrt{2+\sqrt{2}}}\cdot r$	…
Perimetro	$\displaystyle 4\cdot \sqrt{2}\cdot r$	$\displaystyle 8 \cdot \sqrt{2-\sqrt{2}}\cdot r$	$\displaystyle 16 \cdot \sqrt{2-\sqrt{2+\sqrt{2}}}\cdot r$	…
Pi greco	$\displaystyle 2\cdot \sqrt{2}$	$\displaystyle 4 \cdot \sqrt{2-\sqrt{2}}$	$\displaystyle 8\cdot \sqrt{2-\sqrt{2+\sqrt{2}}}$	…
Apotema	$\displaystyle \frac{1}{2}\cdot\sqrt{2}\cdot r$	$\displaystyle \frac{1}{2}\cdot\sqrt{2+\sqrt{2}} \cdot r$	$\displaystyle \frac{1}{2}\cdot\sqrt{2+\sqrt{2+\sqrt{2}}}\cdot r$	…
Delta	$\displaystyle \left(1-\frac{1}{2}\cdot\sqrt{2}\right)\cdot r$	$\displaystyle \left(1-\frac{1}{2}\cdot\sqrt{2+\sqrt{2}}\right) \cdot r$	$\displaystyle \left(1-\frac{1}{2}\cdot\sqrt{2+\sqrt{2+\sqrt{2}}}\right) \cdot r$	…

L’approssimazione di $\pi$ non dipende da $r$ , per $r=1$

Numero lati	4	8	16	32	…
Lato	$\displaystyle \sqrt{2}$	$\displaystyle \sqrt{2-\sqrt{2}}$	$\displaystyle \sqrt{2-\sqrt{2+\sqrt{2}}}$	$\displaystyle \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$	…
Perimetro	$\displaystyle 4\cdot \sqrt{2}$	$\displaystyle 8 \cdot \sqrt{2-\sqrt{2}}$	$\displaystyle 16 \cdot \sqrt{2-\sqrt{2+\sqrt{2}}}$	$\displaystyle 32 \cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$	…
Pi greco	$\displaystyle 2 \cdot \sqrt{2}$	$\displaystyle 4 \cdot \sqrt{2-\sqrt{2}}$	$\displaystyle 8\cdot \sqrt{2-\sqrt{2+\sqrt{2}}}$	$\displaystyle 16\cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$	…
Apotema	$\displaystyle \frac{1}{2}\cdot\sqrt{2}$	$\displaystyle \frac{1}{2}\cdot\sqrt{2+\sqrt{2}}$	$\displaystyle \frac{1}{2}\cdot\sqrt{2+\sqrt{2+\sqrt{2}}}$	$\displaystyle \frac{1}{2}\cdot\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}$	…
Delta	$\displaystyle 1-\frac{1}{2}\cdot\sqrt{2}$	$\displaystyle 1-\frac{1}{2}\cdot\sqrt{2+\sqrt{2}}$	$\displaystyle 1-\frac{1}{2}\cdot\sqrt{2+\sqrt{2+\sqrt{2}}}$	$\displaystyle 1-\frac{1}{2}\cdot\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}$	…

Approssimazioni

Osserva le approssimazioni successive di pi greco e nella colonna a fianco una forma equivalente (produttoria di Viète)

Passo	Numero lati	$\pi$
1	4	$\displaystyle 2 \cdot \sqrt{2}$	$\displaystyle 2 \cdot \frac{2}{\sqrt{2}}$	2,828427124746190
2	8	$\displaystyle 4 \cdot \sqrt{2-\sqrt{2}}$	$\displaystyle 2 \cdot \frac{2}{\sqrt{2}}\cdot \frac{2}{\sqrt{2+\sqrt{2}}}$	3,061467458920718
3	16	$\displaystyle 8 \cdot \sqrt{2-\sqrt{2+\sqrt{2}}}$	$\displaystyle 2 \cdot \frac{2}{\sqrt{2}}\cdot \frac{2}{\sqrt{2+\sqrt{2}}}\cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}$	3,121445152258052
4	32	$\displaystyle 16\cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$	…	3,136548490545939
5	64	$\displaystyle 32\cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$	…	3,1403311569547529
6	128	$\displaystyle 64\cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}$	…	3,1412772509327729
7	256	…	…	3,141513801144301
8	512	…	…	3,141572940367091
9	1024	…	…	3,1415877252771597
10	2048	…	…	3,14159142151119997
…	…	…	…	…

Approfondimenti