pi

	$ n= $
$ pathlength(n)= $
$ \frac{\pi}{2} \approx $	1.5707963267948966

Formula

Let $ n \in \mathbb{N}_{\ge 2} $
Let $ S_2 = [ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}], \quad S_{n+1} = [S_{n_1}, \frac{S_{n_1}+S_{n_{1+1}}}{| S_{n_1}+S_{n_{1+1}} | }, S_{n_2}, \dots, \frac{S_{n_k}+S_{n_{k+1}}}{| S_{n_k}+S_{n_{k+1}} | }, S_{n_k} ] $
Let $$ pathlength(n) = \sum^{Card(S(n)-1)}_{i=1} |S_{n_i+1}-S_{n_i}| $$ $$ \Rightarrow \frac{\pi}{2} = \lim_{n\rightarrow \infty} pathlength(n) $$

Improved forumula

remember: $ \pi = \lim_{n\rightarrow \infty} n\cdot sin(\frac{180°}{n}) $ Archimedes formula for pi We can therefor deduce, that $$pathlength (n) = Card(S(n)-1) \cdot | S_{n_1}-S_{n_0} |$$ And therefor $$ |S_{n_{i+1}} - S_{n_i}| = |S_{n_{j+1}} - S_{n_j}|, \quad 0 < i,j < Card(S(n)-1) $$

	\( n= \)
\( pathlength(n)= \)
\( \frac{\pi}{2} \approx \)	1.5707963267948966