Das regelmäßige Zwanzigeck in SVG

Das regelmäßige Zwanzigeck hat zwanzig gleich lange Seiten, die an jedem Eckpunkt einen Winkel von $162^{o}={\frac {9\pi }{10}}$ einschließen.

Viele Aussagen über das regelmäßige regelmäßige Fünfeck gelten auch für das regelmäßige Zehneck und das regelmäßige Zwanzigeck. Der Goldene Schnitt $\varphi =$${\frac {1}{2}}({\sqrt {5}}+1)$ und die Zahl ${\sqrt {5}}$ spielen eine wichtige Rolle.

Anm.: die externen Links führen meist zu Wolfram Alpha mit der Eingabe der entsprechenden Formel.

Eckpunkte

Die $n$ Eckpunkte eines regelmäßigen N-Ecks mit Umkreisradius $R$ ergeben sich bekanntlich als

$P_{k}=\left(R\cos {\frac {k\pi }{n}},R\sin {\frac {k\pi }{n}}\right)\quad k=0,\ldots ,n-1$

Zunächst definiere ich $\varphi =$${\frac {1}{2}}({\sqrt {5}}+1)$ und $\Xi ={\sqrt {1-{\frac {\varphi ^{2}}{4}}}}={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$

$\sin 36^{o}$$=\Xi ={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},$ $\cos 36^{o}={\frac {1}{4}}({\sqrt {5}}+1)={\frac {\varphi }{2}}$

$\sin 54^{o}=\cos 36^{o}={\frac {\varphi }{2}}={\frac {1}{4}}({\sqrt {5}}+1)$, $\cos 54^{o}=\sin 36^{o}={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$

$\sin 72^{o}=\Xi *\varphi ={\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}$, $\cos 72^{o}={\frac {1}{2\varphi }}$

Eckpunkte im 1. Quadranten

Winkel	cos	sin	Eckpunkt	R=754
$0^{o}$	$1$	$0$	$(1,0)$	(754,0)
$18^{o}={\frac {\pi }{10}}$	$\Xi *\varphi$	${\frac {1}{2\varphi }}$	$\left({\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}},{\frac {1}{4}}({\sqrt {5}}-1)\right)$	(717.1,233)
$36^{o}={\frac {\pi }{5}}$	${\frac {\varphi }{2}}$	$\Xi$	$\left({\frac {1}{4}}(1+{\sqrt {5}}),{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(610,443.19)
$54^{o}={\frac {3\pi }{10}}$	$\Xi$	${\frac {\varphi }{2}}$	$\left(({\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},{\frac {1}{4}}(1+{\sqrt {5}})\right)$	(443.19,610)
$72^{o}={\frac {2\pi }{5}}$	${\frac {1}{2\varphi }}$	$\Xi *\varphi$	$\left({\frac {1}{4}}({\sqrt {5}}-1),{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(233,717.1)
$90^{o}={\frac {\pi }{4}}$	$0$	$1$	$\left(0,1\right)$	(0,754)

Pfad in SVG

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   d="m754,0 L717.1,233 610,443.19 443.19,610 233,717.1 0,754
   -233,717.1 -443.19,610 -610,443.19 -717.1,233 -754,0
   -717.1,-233 -610,-443.19 -443.19,-610 -233,-717.1 0,-754
   233,-717.1 443.19,-610 610,-443.19 717.1,-233
   z" />

Sinus/Cosinus von 36° und 72°

Winkelfunktion	Radius	377	610	233	131	754
$\sin 36^{o}=\sin {\frac {\pi }{5}}$	0,809016994	304,9994069	493,5003666	188,5009597	105,9812263	609,9988138
$\cos 36^{o}=\cos {\frac {\pi }{5}}$	0,587785252	221,5950401	358,5490039	136,9539638	76,99986805	443,1900802
$\sin 72^{o}=\sin {\frac {2\pi }{5}}$	0,309016994	116,4994069	188,5003666	72,00095969	40,48122626	232,9988138
$\cos 72^{o}=\cos {\frac {2\pi }{5}}$	0,951056516	358,5483066	580,1444749	221,5961683	124,5884036	717,0966133

Zahlentabelle Goldener Schnitt

Zahlenwert	ungefähr	Kettenbruch	sin	cos	Näherung
${\frac {1}{2}}(1+{\sqrt {5}})=\Phi$	$\approx 1.61803398875$	$[1;{\overline {1}}]$	$1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,\ldots$ ${\frac {8}{5}}\diamond {\frac {13}{8}}\diamond {\frac {21}{13}}\diamond {\frac {34}{21}}\diamond \ldots$
${\frac {1}{2}}({\sqrt {5}}-1)=\Phi -1$ $={\frac {1}{\Phi }}$	$\approx 0.61803398875$	$[0;{\overline {1}}]$	${\frac {5}{8}}\diamond {\frac {8}{13}}\diamond {\frac {13}{21}}\diamond {\frac {21}{34}}\diamond \ldots$
${\frac {1}{4}}({\sqrt {5}}-1)={\frac {\Phi -1}{2}}$ $={\frac {1}{2\Phi }}$	$\approx 0.3090169947$	$[0;3,{\overline {4}}]$	$18^{o}={\frac {\pi }{10}}$	$72^{o}={\frac {2\pi }{5}}$	${\frac {4}{13}}\diamond {\frac {17}{55}}\diamond {\frac {72}{233}}$
${\frac {\sqrt {2}}{4}}{\sqrt {5-{\sqrt {5}}}}$ $={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$ $={\frac {1}{2}}{\sqrt {{\frac {1}{2}}(5-{\sqrt {5}})}}$	$\approx 0.58778525229$	$[0;1,1,2,2,1,6,1,56,\ldots ]$	$36^{o}={\frac {\pi }{5}}$	$54^{o}={\frac {3\pi }{10}}$	${\frac {4379}{7450}}\diamond {\frac {77}{131}}$
${\frac {1}{4}}(1+{\sqrt {5}})={\frac {\Phi }{2}}$	$\approx 0.8090169943749474$	$[0;1,{\overline {4}}]$	$54^{o}={\frac {3\pi }{10}}$	$36^{o}={\frac {\pi }{5}}$	${\frac {305}{377}}\diamond {\frac {72}{89}}\diamond {\frac {17}{21}}$
${\frac {\sqrt {2}}{4}}{\sqrt {5+{\sqrt {5}}}}$ $={\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}$ $={\frac {1}{2}}{\sqrt {{\frac {1}{2}}(5+{\sqrt {5}})}}$	$\approx 0.9510565162951536$	$[0;1,19,2,3,6,5,1,1,1,\ldots ]$	$72^{o}={\frac {2\pi }{5}}$	$18^{o}={\frac {\pi }{10}}$	${\frac {855}{899}}\diamond {\frac {136}{143}}\diamond {\frac {39}{41}}$
			$108^{o}={\frac {3\pi }{5}}$	$-162^{o}=-{\frac {9\pi }{10}}$
${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ $={\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}$	$\approx 0.2588190$	$[0;3,1,6,2,1,30,5,2,9,3,1,\ldots ]$	$15^{o}={\frac {\pi }{12}}$	$75^{o}={\frac {5\pi }{12}}$	${\frac {22}{85}}\diamond {\frac {675}{2608}}$
${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ $={\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}$	$\approx 0.96592582$	$[0;1,28,2,1,7,21,1,8,1,\ldots ]$	$75^{o}={\frac {5\pi }{12}}$	$15^{o}={\frac {\pi }{12}}$	${\frac {652}{675}}\diamond {\frac {85}{88}}$
${\frac {\sqrt {2}}{2}}$	$\approx 0.7071067811865475$	$[0;1,{\overline {2}}]$	$45^{o}={\frac {\pi }{4}}$		${\frac {5}{7}}\diamond {\frac {12}{17}}\diamond {\frac {29}{41}}\diamond {\frac {70}{99}}\diamond {\frac {169}{239}}\diamond {\frac {408}{577}}$
${\frac {1}{2}}$			$30^{o}={\frac {\pi }{6}}$	$60^{o}={\frac {\pi }{3}}$
${\frac {\sqrt {3}}{2}}$	$\approx 0.8660254037844386$	$[0;1,{\overline {6,2}}]$	$60^{o}={\frac {\pi }{3}}$	$30^{o}={\frac {\pi }{6}}$	${\frac {84}{97}}\diamond {\frac {181}{209}}\diamond {\frac {1170}{1351}}$
${\sqrt {2}}$	$\approx 1.414213562373095$	$[1;{\overline {2}}]$	${\frac {99}{70}}\diamond {\frac {239}{169}}\diamond {\frac {577}{408}}$
${\sqrt {3}}$	$\approx 1.732050807568877$	$[1;{\overline {1,2}}]$	${\frac {97}{56}}\diamond {\frac {265}{153}}\diamond {\frac {362}{209}}\diamond {\frac {1351}{780}}$
${\sqrt {5}}$	$\approx 2.23606797749979$	$[2;{\overline {4}}]$	${\frac {9}{4}}\diamond {\frac {38}{17}}\diamond {\frac {161}{72}}\diamond {\frac {682}{305}}$
$4{\frac {({\sqrt {2}}-1)}{3}}=\varkappa$	$\approx 0.5522847498307933$	$[0;{\overline {1,1,4,3}}]$	${\frac {5}{9}}\diamond {\frac {16}{29}}\diamond {\frac {21}{38}}\diamond {\frac {37}{67}}\diamond {\frac {169}{306}}\diamond {\frac {544}{985}}$

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