User:Lilyuserin/SVG/kite+dart

Allgemeines

Bei Drachen (kite) und Pfeil (dart) handelt es sich um Vierecke mit exakt einer Spiegelachse und mit zwei Paaren von gleichlangen Seiten. Das Verhältnis von langer zu kurzer Seite entspricht dem Goldenen Schnitts $\varphi =$${\frac {1}{2}}({\sqrt {5}}+1)$ zueinander stehen. Die Drachen genannte Figur ist eine konvexe Form, der Pfeil hingegen hat die kurzen Seiten nach Innen geklappt. Beim Drachen sind drei Innenwinkel 72° und der zwischen den beiden kurzen Seiten 144°. Beim Pfeil ist der Winkel zwischen den langen Seiten 72°, der Außenwinkel zwischen den beiden kurzen Seiten 144° und die beiden übrigen Innenwinkel 36°.

Mit diesen Formen, den sogenannten Penrose-Kacheln kann die Ebene lückenlos bedeckt werden, ohne dass eine Wiederholung auftritt, die sogenannte Penrose-Parkettierung. Eine weitere Kombination sind zwei Rauten, die dicke (Eckwinkel: 72° und 108°) und die dünne Raute (Eckwinkel 36° und 144°).

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

Eckpunkte

$\varphi =$${\frac {1}{2}}({\sqrt {5}}+1)$

$\Xi ={\sqrt {1-{\frac {\varphi ^{2}}{4}}}}={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$

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

$\sin 144^{o}$$=\Xi ={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},$ $\cos 144^{o}=-{\frac {1}{4}}({\sqrt {5}}+1)=-{\frac {\varphi }{2}}$

Drache

Eckpunkte

Eckpunkt	Koordinaten	Formeln	Zahlenwerte
$A$	$(0,0)$	Einheitswert: 754
$B$	$(\varphi \cos 36^{0},-\varphi \sin 36^{0})=\left({\frac {\varphi ^{2}}{2}},-\varphi *\Xi \right)$	$({\frac {3+{\sqrt {5}}}{4}}$, $-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}})$	(987,-717.1)
$C$	$(\varphi ,0)$	$\left({\frac {1}{2}}({\sqrt {5}}+1),0\right)$	(1220,0)
$D$	$(\varphi \cos 36^{0},\varphi \sin 36^{0})=\left({\frac {\varphi ^{2}}{2}},\varphi *\Xi \right)$	$\left({\frac {3+{\sqrt {5}}}{4}},{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(987,717.1)

Pfad

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   d="m1220,0 L987,-717.1 0,0 987,717.1 z"/>

Relativkoordinaten:

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   d="m1220,720 -233,-717.1 -987 717.1 987,717.1 z" />

Pfeil

Eckpunkte

Eckpunkt	Koordinaten	Formeln	Zahlenwerte
$A$	$(0,0)$	Einheitswert: 754
$B$	$(\varphi \cos 36^{0},-\varphi \sin 36^{0})=\left({\frac {\varphi ^{2}}{2}},-\varphi *\Xi \right)$	$({\frac {3+{\sqrt {5}}}{4}}$, $-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}})$	(987,-717.1)
$C$	$(1,0)$	(1,0)	(754,0)
$D$	$(\varphi \cos 36^{0},\varphi \sin 36^{0})=\left({\frac {\varphi ^{2}}{2}},\varphi *\Xi \right)$	$\left({\frac {3+{\sqrt {5}}}{4}},{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(987,717.1)

Pfad

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   d="m754,0 L987,-717.1 0,0 987,717.1 z" />

Relativkoordinaten

<path
   d="m754,717 233,-717.1 -987,717.1 987,717.1 z" />

Dicke Raute

Eckpunkte

Eckpunkt	Koordinaten	Formeln	Zahlenwerte
$A$	$(0,0)$	Einheitswert: 754
$B$	$(\varphi \cos 36^{0},-\varphi \sin 36^{0})=\left({\frac {\varphi ^{2}}{2}},-\varphi *\Xi \right)$	$({\frac {3+{\sqrt {5}}}{4}}$, $-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}})$	(987,-717.1)
$C$	$\left(2\varphi \cos 36^{0},0\right)=\left(2\varphi ^{2},0\right)$	$\left({\frac {3+{\sqrt {5}}}{2}},0\right)$	(1974,0)
$D$	$(\varphi \cos 36^{0},\varphi \sin 36^{0})=\left({\frac {\varphi ^{2}}{2}},\varphi *\Xi \right)$	$\left({\frac {3+{\sqrt {5}}}{4}},{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(987,717.1)

Pfad

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Relativkoordinaten

<path
   d="m0,713 987,717.1 987,-717 -987,-717.1 z" />

Dünne Raute

Eckpunkte

Eckpunkt	Koordinaten	Formeln	Zahlenwerte
$A$	$(0,0)$	Einheitswert: 754
$B$	$({\frac {1}{2\varphi }},-\varphi \sin 36^{0})=\left({\frac {\varphi -1}{2}},-\varphi *\Xi \right)$	$({\frac {1}{1+{\sqrt {5}}}}$, $-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}})$	(233,-717.1)
$C$	$\left({\frac {1}{\varphi }},0\right)=\left(\varphi -1,0\right)$	$\left({\frac {2}{{\sqrt {5}}+1}},0\right)$	(466,0)
$D$	$({\frac {1}{2\varphi }},\varphi \sin 36^{0})=\left({\frac {\varphi -1}{2}},\varphi *\Xi \right)$	$({\frac {1}{1+{\sqrt {5}}}}$, ${\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}})$	(233,717.1)

Pfad

<path
   d="m466,0 L233,717.1 0,0 233,-717.1 z" />

Relativkoordinaten

<path
   d="m466 717.1 -233,717.1 -233,-717.1 233,-717.1 z" />

Galerie

• 
  Drache
• 
  Pfeil
• 
  dicke Raute
• 
  dünne Raute
• 
  dicke+dünne Raute

Sinus/Cosinus von 36° und 72°

	Radius	377	754
Winkelfunktion
$\sin 36^{o}=\sin {\frac {\pi }{5}}$	0,809016994	304,9994069	609,9988138
$\cos 36^{o}=\cos {\frac {\pi }{5}}$	0,587785252	221,5950401	443,1900802
$\sin 72^{o}=\sin {\frac {2\pi }{5}}$	0,309016994	116,4994069	232,9988138
$\cos 72^{o}=\cos {\frac {2\pi }{5}}$	0,951056516	358,5483066	717,0966133

Annäherung durch Bruchzahlen

Tabelle • Tabellenseite bearbeiten

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

Links

• Decagon
• Zehneck
• 10fach Rotationssymmetrie
• Pentagon
• Pentagon
• Fünfeck