Dresden Algorithmics Channel: Module 35

The complex function f with the real-valued argument x, the parameter n, and the real weight function $\varphi$ is defined by a nested expression

f (n, $\displaystyle \varphi$ , x)	=	e^2(0)x j $\displaystyle \left(\vphantom{1+e^{2\varphi(1)\pi x\,j}\left(1+e^{2\varphi(2)\p... ...left(1+\ldots+e^{2\varphi(n)\pi x\,j}\left(1\right)\ldots\right)\right)}\right.$ 1 + e^2(1)x j $\displaystyle \left(\vphantom{1+e^{2\varphi(2)\pi x\,j}\left(1+\ldots+e^{2\varphi(n)\pi x\,j}\left(1\right)\ldots\right)}\right.$ 1 + e^2(2)x j $\displaystyle \left(\vphantom{1+\ldots+e^{2\varphi(n)\pi x\,j}\left(1\right)\ldots}\right.$ 1 +...+ e^2(n)x j $\displaystyle \left(\vphantom{1}\right.$ 1 $\displaystyle \left.\vphantom{1}\right)$ ... $\displaystyle \left.\vphantom{1+\ldots+e^{2\varphi(n)\pi x\,j}\left(1\right)\ldots}\right)$ $\displaystyle \left.\vphantom{1+e^{2\varphi(2)\pi x\,j}\left(1+\ldots+e^{2\varphi(n)\pi x\,j}\left(1\right)\ldots\right)}\right)$ $\displaystyle \left.\vphantom{1+e^{2\varphi(1)\pi x\,j}\left(1+e^{2\varphi(2)\p... ...left(1+\ldots+e^{2\varphi(n)\pi x\,j}\left(1\right)\ldots\right)\right)}\right)$ ;	(1)
		n $\displaystyle \in$ $\mathbb{N}$ ₀ ;
		$\displaystyle \varphi$ (i) $\displaystyle \in$ $\displaystyle \left\{\vphantom{i\,,\,\frac{1}{2}\,i(i+1)\,,\,\frac{1}{6}\,i(i+1... ...,\,\mbox{fib}(i) (i--te Fibonacci--Zahl)\,, p_i (i--te Primzahl),\ldots}\right.$ i , $\displaystyle {\textstyle\frac{1}{2}}$ i(i + 1) , $\displaystyle {\textstyle\frac{1}{6}}$ i(i + 1)(i + 2) , fib(i) (ith Fibonacci number) , p_i (ith prime number) , ... $\displaystyle \left.\vphantom{i\,,\,\frac{1}{2}\,i(i+1)\,,\,\frac{1}{6}\,i(i+1)... ...\,\mbox{fib}(i) (i--te Fibonacci--Zahl)\,, p_i (i--te Primzahl),\ldots}\right\}$ ;
		x $\displaystyle \in$ $\mathbb{R}$ ; j = $\displaystyle \sqrt{-1}$ ; f (n, $\displaystyle \varphi$ , x) $\displaystyle \in$ $\mathbb{C}$

and generates for n = 0, 1, 2,... the sequence of complex numbers

$\begin{displaymath}<f(n,\varphi,x)>_{n=0}^{\mbox{\scriptsize ad infinitum}}. \end{displaymath}$

(2)

These complex numbers can be represented as points in the Gauss plane:

$\epsfbox{modul35_Figure1_1}$
Figure 1: n = 900 , $\varphi$ (i) = i , x = $\pi$
$\pi$ spiral

$\epsfbox{modul35_Figure1}$
Figure 2: n = 600 , $\varphi$ (i) = i , x = ${\frac{\pi}{2}}$ ;
$\pi$ sea horse

$\epsfbox{modul35_Figure7}$
Figure 3: n = 600 , $\varphi$ (i) = i , x = ${\frac{\pi}{4}}$

$\epsfbox{modul35_Figure8}$
Figure 4 : n = 600 , $\varphi$ (i) = i , x = e

$\epsfbox{modul35_Figure4}$
Figure 5: n = 300 , $\varphi$ (i) = i , x = ${\frac{1}{33}}$

$\epsfbox{modul35_Figure5_1}$
Figure 6: n = 165 , $\varphi$ (i) = i , x = ${\frac{1}{163}}$
M51

$\epsfbox{modul35_Figure5}$
Figure 7: n = 490 , $\varphi$ (i) = i , x = ${\frac{1}{163}}$

$\epsfbox{modul35_Figure6}$
Figure 8: n = 800 , $\varphi$ (i) = i , x = ${\frac{1}{\pi}}$

$\epsfbox{modul35_Figure2}$
Figure 9: n = 500 , $\varphi$ (i) = i , x = $\sqrt{163}$

$\epsfbox{modul35_Figure3}$
Figure 10: n = 500 , $\varphi$ (i) = i , x = ${\frac{e}{\pi}}$

$\epsfbox{modul35_Figure9}$
Figure 11: n = 400 , $\varphi$ (i) = i,
x = $\delta$ = 4.66920160910299...

$\epsfbox{modul35_Figure9_1}$
Figure 12: n = 500 , $\varphi$ (i) = i , x = ${\frac{1}{\delta}}$

$\epsfbox{modul35_Figure10}$
Figure 13: n = 400 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{128}}$

$\epsfbox{modul35_Figure11}$
Figure 14: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{256}}$

$\epsfbox{modul35_Figure12_1}$
Figure 15: n = 262 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{\delta}{2}}$

$\epsfbox{modul35_Figure12}$
Figure 16: n = 300 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = $\pi$ ;
$\pi$ cats

$\epsfbox{modul35_Figure13}$
Figure 17: n = 300 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{43}}$

$\epsfbox{modul35_Figure1_14}$
Figure 18: n = 300 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{67}}$
stagbeetle clones

$\epsfbox{modul35_Figure15}$
Figure 19: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{139}}$

$\epsfbox{modul35_Figure16}$
Figure 20: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{163}}$
ant and ant and ... or?

$\epsfbox{modul35_Figure17}$
Figure 21: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{173}}$

$\epsfbox{modul35_Figure18}$
Figure 22: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{179}}$

$\epsfbox{modul35_Figure19}$
Figure 23: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{181}}$

$\epsfbox{modul35_Figure20}$
Figure 24: n = 600 , $\varphi$ (i) = ${\frac{1}{2}}$ i(i + 1) , x = ${\frac{1}{191}}$

Module 35

1st part

Curlicue variations

Polygon patterns in the Gauss plane of complex numbers