Module 36

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The Feigenbaum-constant δ in the Gauss plane

DACModule 35 shows:

The function f 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) $\displaystyle \in$ $\mathbb{C}$

Representing the values of the function by points in the Gauss plane

pset(f (n, $\displaystyle \varphi$ , x)) = pset(re(f (n, $\displaystyle \varphi$ , x)), im(f (n, $\displaystyle \varphi$ , x)))

(2)

and connecting adjacent points by straight lines generates according to algorithm (A1) Curlicue-like fractal polygon patterns.

(A1)

With certain values for x as well as suitable weighting functions $\varphi$ (i) we obtain characteristic closed-form or repetitive fractal patterns and pattern elements that resemble the "inner structure" of x. Yet, an interpretation of the process needs to be developed. Note the similarity to continued fraction representation see DACModule 16.
We observe a special aesthetic appeal in the interesting ornamental character of some of the generated patterns and pattern elements.

The Feigenbaum-constant (see DACModule 24)

$\displaystyle \delta$ = 4.669201609102990671853203820466201617258185577475768632745651...

(3)

is such a pictogene mathematical constant which we will explore in the following.

$\fbox{$\varphi(i)=i$ }$

$\epsfbox{delta}$
Figure 1: n = 800 , x = $\delta$ $\epsfbox{delta_bluete}$
Figure 2: n = 130 , x = $\delta$ , rosette

$\epsfbox{delta_2000}$
Figure 3: n = 2000 , x = $\delta$ $\epsfbox{zwei_delta}$
Figure 4: n = 800 , x = 2 $\delta$

$\epsfbox{drei_delta}$
Figure 5: n = 800 , x = 3 $\delta$ $\epsfbox{drei_delta_bluete}$
Figure 6: n = 130 , x = 3 $\delta$ , rosette

$\epsfbox{vier_delta}$
Figure 7: n = 800 , x = 4 $\delta$ $\epsfbox{fuenf_delta}$
Figure 8: n = 800 , x = 5 $\delta$

$\epsfbox{sechs_delta}$
Figure 9: n = 800 , x = 6 $\delta$ $\epsfbox{sieben_delta}$
Figure 10: n = 800 , x = 7 $\delta$

$\epsfbox{acht_delta}$
Figure 11: n = 800 , x = 8 $\delta$ $\epsfbox{neun_delta}$
Figure 12: n = 1400 , x = 9 $\delta$

$\epsfbox{zehn_delta}$
Figure 13: n = 1400 , x = 10 $\delta$ $\epsfbox{elf_delta}$
Figure 14: n = 1400 , x = 11 $\delta$

$\epsfbox{zwoelf_delta}$
Figure 15: n = 1200 , x = 12 $\delta$ $\epsfbox{fuenfzehn_delta}$
Figure 16: n = 2000 , x = 15 $\delta$

$\epsfbox{delta_halbe}$
Figure 17: n = 790 , x = ${\frac{\delta}{2}}$ $\epsfbox{delta_halbe_bluete}$
Figure 18: n = 260 , x = ${\frac{\delta}{2}}$ , rosette

$\epsfbox{delta_drittel}$
Figure 19: n = 1400 , x = ${\frac{\delta}{3}}$ $\epsfbox{delta_viertel}$
Figure 20: n = 800 , x = ${\frac{\delta}{4}}$

$\epsfbox{delta_teilt_5}$
Figure 21: n = 1400 , x = ${\frac{\delta}{5}}$ $\epsfbox{delta_teilt_6}$
Figure 22: n = 1400 , x = ${\frac{\delta}{6}}$

$\epsfbox{delta_teilt_7}$
Figure 23: n = 2000 , x = ${\frac{\delta}{7}}$ $\epsfbox{delta_teilt_7_bluete}$
Figure 24: n = 900 , x = ${\frac{\delta}{7}}$ , rosette

$\epsfbox{delta_quadr}$
Figure 25: n = 1400 , x = $\delta^{2}_{}$ $\epsfbox{delta_quadr_bluete}$
Figure 26: detail of figure 25:
k = 141...413 , x = $\delta^{2}_{}$

$\epsfbox{delta_wurzel}$
Figure 27: n = 800 , x = $\sqrt{\delta}$
(see DACModul 35, figures 4 and 10) $\epsfbox{e_hoch_delta}$
Figure 28: n = 400 , x = e

$\epsfbox{1_2durchdelta}$
Figure 29: n = 800 , x = 1 - ${\frac{2}{\delta}}$
(see DACModul 35, figure 2 and below figures 30 - 39)

$\epsfbox{wurzel2_modif}$
Figure 30: n = 1400 , x = ${\frac{2^{\sqrt{2}}}{2+2^{\sqrt{2}}}}$ $\epsfbox{wurzel2_modif_bluete}$
Figure 31: n = 990 , x = ${\frac{2^{\sqrt{2}}}{2+2^{\sqrt{2}}}}$ , rosette

$\epsfbox{log}$
Figure 32: n = 1000 , x = ln(1 + ${\frac{2\pi}{3e}}$ ) $\epsfbox{10_w163}$
Figure 33: n = 1000 , x = ${\frac{150}{\sqrt{163}}}$

$\epsfbox{4_teilt_7}$
Figure 34: n = 100 , x = ${\frac{4}{7}}$ $\epsfbox{2_hoch}$
Figure 35: n = 1400 , x = 2 - 1

$\epsfbox{modif_gold_schnitt}$
Figure 36: n = 1000 , x = 15 (2 $\sqrt{\frac{1}{2}(\sqrt{5}-1)}$ - 1) $\epsfbox{modif_gold_schnitt2}$
Figure 37: n = 1000 , x = 2 $\sqrt{\frac{1}{2}(\sqrt{5}-1)}$

$\epsfbox{pi_2e}$
Figure 38: n = 1300 , x = ${\frac{15\,\pi}{2\,e}}$ $\epsfbox{pi_2e_ohne}$
Figure 39: n = 1300 , x = ${\frac{15\,\pi}{2\,e}}$
(set of points, without polygon line)

$\fbox{$\varphi(i)=\lceil\sqrt{i}\rceil$ }$
$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_wurzel_delta}\\ {\small Bild 40: \em $n=1000\,,\;x=\delta$ }}$
$\fbox{$\varphi(i)=\lfloor \frac{i}{\ln (i+3)}\rfloor$ }$
$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_log_delta}\\ {\small Bild 41: \em $n=1000\,,\;x=\delta$ }}$

$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_log_delta2}\\ {\small Bild 42: \em $k=320\ldots 440\,,\;x=\delta$ }}$ $\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_log_delta3}\\ {\small Bild 43: \em $k=610\ldots 780\,,\;x=\delta$ }}$

$\fbox{$\varphi(i)=(7i+41)\mbox{mod} 1023$ }$
$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_mod1023_delta}\\ {\small Bild 44: \em $n=2000\,,\;x=\delta$ }}$
$\fbox{$\varphi(i)=(7i+41)\mbox{mod} 163$ }$
$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_mod_delta}\\ {\small Bild 45: \em $n=2000\,,\;x=\delta$ }}$

$\fbox{$\varphi(i)=i^{11}\mbox{mod} 127$ }$
$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_mod127_hoch_delta}\\ {\small Bild 46: \em $n=2000\,,\;x=\delta$ }}$
$\fbox{$\varphi(i)=i^{3}\mbox{mod} 127$ }$
$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_mod127_hoch3_delta}\\ {\small Bild 47: \em $n=2000\,,\;x=\delta$ }}$
$\fbox{$\varphi(i)=i^{3}\mbox{mod} 63$ }$

$\textstyle \parbox{7.5cm}{\epsfxsize=6.cm \epsfbox{samml_mod63_hoch3_delta}\\ {\small Bild 48: \em $n=2000\,,\;x=\delta$ }}$

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