User:Lilyuserin/SVG/pentagon

Allgemeines

Das regelmäßige Fünfeck hat fünf gleich lange Seiten, die an jedem Eckpunkt einen Winkel von $108^{o}={\frac {3\pi }{5}}$ einschließen. Figuren mit ähnlichen Eigenschaften sind das Zehneck und das 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 72^{o}=\Xi *\varphi ={\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}

,

\cos 72^{o}={\frac {1}{2\varphi }}

\sin 144^{o}

=\Xi ={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},

\cos 144^{o}=-{\frac {1}{4}}({\sqrt {5}}+1)=-{\frac {\varphi }{2}}

Eckpunkte tabellarisch

Aus den Koordinaten für das regelmäßige Zehneck lässt sich leicht ein Fünfeck ableiten, indem man jeden zweiten Eckpunkt weglässt. Aus dem hier beschriebenen Zehneck kann man kein Fünfeck erzeugen, dessen Basis parallel zur X-Achse liegt, man muss es um $\pm 18^{o}$ drehen.

Winkel	$cos$	$sin$	Eckpunkt	R=754
$0^{o}$	$1$	$0$	$(1,0)$	(754,0)
$72^{o}={\frac {2\pi }{5}}$	${\frac {1}{2\varphi }}$	$\Xi *\varphi$	$\left({\frac {1}{4}}({\sqrt {5}}-1),{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(233,717.1)
$144^{o}={\frac {4\pi }{5}}$	$-{\frac {\varphi }{2}}$	$\Xi$	$\left(-{\frac {1}{4}}(1+{\sqrt {5}}),{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(-610,443.19)
$216^{o}={\frac {6\pi }{5}}$	$-{\frac {\varphi }{2}}$	$-\Xi$	$\left(-{\frac {1}{4}}(1+{\sqrt {5}}),-{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(-610,-443.19)
$288^{o}={\frac {8\pi }{5}}$	${\frac {1}{2\varphi }}$	$-\Xi *\varphi$	$\left({\frac {1}{4}}({\sqrt {5}}-1),-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(233,-717.1)

Pfad in SVG

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Zahlenwerte

Tabelle • Tabellenseite bearbeiten

Zahlenwert	ungefähr	Kettenbruch	sin	cos	Näherung
${\frac {1}{2}}(1+{\sqrt {5}})=\Phi$	$\approx 1.61803398875$	$[1;{\overline {1}}]$	$1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,\ldots$ ${\frac {8}{5}}\diamond {\frac {13}{8}}\diamond {\frac {21}{13}}\diamond {\frac {34}{21}}\diamond \ldots$
${\frac {1}{2}}({\sqrt {5}}-1)=\Phi -1$ $={\frac {1}{\Phi }}$	$\approx 0.61803398875$	$[0;{\overline {1}}]$	${\frac {5}{8}}\diamond {\frac {8}{13}}\diamond {\frac {13}{21}}\diamond {\frac {21}{34}}\diamond \ldots$
${\frac {1}{4}}({\sqrt {5}}-1)={\frac {\Phi -1}{2}}$ $={\frac {1}{2\Phi }}$	$\approx 0.3090169947$	$[0;3,{\overline {4}}]$	$18^{o}={\frac {\pi }{10}}$	$72^{o}={\frac {2\pi }{5}}$	${\frac {4}{13}}\diamond {\frac {17}{55}}\diamond {\frac {72}{233}}$
${\frac {\sqrt {2}}{4}}{\sqrt {5-{\sqrt {5}}}}$ $={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$ $={\frac {1}{2}}{\sqrt {{\frac {1}{2}}(5-{\sqrt {5}})}}$	$\approx 0.58778525229$	$[0;1,1,2,2,1,6,1,56,\ldots ]$	$36^{o}={\frac {\pi }{5}}$	$54^{o}={\frac {3\pi }{10}}$	${\frac {4379}{7450}}\diamond {\frac {77}{131}}$
${\frac {1}{4}}(1+{\sqrt {5}})={\frac {\Phi }{2}}$	$\approx 0.8090169943749474$	$[0;1,{\overline {4}}]$	$54^{o}={\frac {3\pi }{10}}$	$36^{o}={\frac {\pi }{5}}$	${\frac {305}{377}}\diamond {\frac {72}{89}}\diamond {\frac {17}{21}}$
${\frac {\sqrt {2}}{4}}{\sqrt {5+{\sqrt {5}}}}$ $={\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}$ $={\frac {1}{2}}{\sqrt {{\frac {1}{2}}(5+{\sqrt {5}})}}$	$\approx 0.9510565162951536$	$[0;1,19,2,3,6,5,1,1,1,\ldots ]$	$72^{o}={\frac {2\pi }{5}}$	$18^{o}={\frac {\pi }{10}}$	${\frac {855}{899}}\diamond {\frac {136}{143}}\diamond {\frac {39}{41}}$
	$\approx 0.9510565162951536$	$[0;1,19,2,3,6,5,1,1,1,\ldots ]$	$108^{o}={\frac {3\pi }{5}}$	$-162^{o}=-{\frac {9\pi }{10}}$
${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ $={\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}$	$\approx 0.2588190$	$[0;3,1,6,2,1,30,5,2,9,3,1,\ldots ]$	$15^{o}={\frac {\pi }{12}}$	$75^{o}={\frac {5\pi }{12}}$	${\frac {22}{85}}\diamond {\frac {675}{2608}}$
${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ $={\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}$	$\approx 0.96592582$	$[0;1,28,2,1,7,21,1,8,1,\ldots ]$	$75^{o}={\frac {5\pi }{12}}$	$15^{o}={\frac {\pi }{12}}$	${\frac {652}{675}}\diamond {\frac {85}{88}}$
${\frac {\sqrt {2}}{2}}$	$\approx 0.7071067811865475$	$[0;1,{\overline {2}}]$	$45^{o}={\frac {\pi }{4}}$		${\frac {5}{7}}\diamond {\frac {12}{17}}\diamond {\frac {29}{41}}\diamond {\frac {70}{99}}\diamond {\frac {169}{239}}\diamond {\frac {408}{577}}$
${\frac {1}{2}}$			$30^{o}={\frac {\pi }{6}}$	$60^{o}={\frac {\pi }{3}}$
${\frac {\sqrt {3}}{2}}$	$\approx 0.8660254037844386$	$[0;1,{\overline {6,2}}]$	$60^{o}={\frac {\pi }{3}}$	$30^{o}={\frac {\pi }{6}}$	${\frac {84}{97}}\diamond {\frac {181}{209}}\diamond {\frac {1170}{1351}}$
${\sqrt {2}}$	$\approx 1.414213562373095$	$[1;{\overline {2}}]$	${\frac {99}{70}}\diamond {\frac {239}{169}}\diamond {\frac {577}{408}}$
${\sqrt {3}}$	$\approx 1.732050807568877$	$[1;{\overline {1,2}}]$	${\frac {99}{70}}\diamond {\frac {265}{153}}\diamond {\frac {362}{209}}\diamond {\frac {1351}{780}}$
${\sqrt {5}}$	$\approx 2.23606797749979$	$[2;{\overline {4}}]$	${\frac {95}{56}}\diamond {\frac {521}{233}}\diamond {\frac {682}{305}}$
$4{\frac {({\sqrt {2}}-1)}{3}}=\varkappa$	$\approx 0.5522847498307933$	$[0;{\overline {1,1,4,3}}]$	${\frac {5}{9}}\diamond {\frac {16}{29}}\diamond {\frac {21}{38}}\diamond {\frac {37}{67}}\diamond {\frac {169}{306}}\diamond {\frac {544}{985}}$

Galerie

Pentagon, regelmäßiges Fünfeck
Decagon, regelmäßiges Zehneck
Icosagon, regelmäßiges Zwanzigeck

SVG-Code

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