File:01-257-Eck E-15.svg

Summary[edit]

Description01-257-Eck E-15.svg	Deutsch: 257-Eck, Näherungskonstruktion der ersten Seite English: 257-gon, approximate construction of the first side
Date	19 April 2016
Source	Own work
Author	Petrus3743
Other versions
SVG development InfoField	The SVG code is valid.   This trigonometry was created with GeoGebra by Petrus3743.   This SVG trigonometry uses the path text method.

Näherungsonstruktion[edit]

• Die Methode ähnelt der des Dreizehnecks

Ergebnis[edit]

Bezogen auf den Einheitskreis r = 1 [LE]

• Konstruierte Seite des 257-Ecks in GeoGebra (Anzeige 15 signifikante Nachkommastellen, gerundet) ${\displaystyle {\overline {E_{257}E_{1}}}=a=2{,}44475829850863E-2\;[LE]}$
• Seite des 257-Ecks, 15 signifikante Nachkommastellen, ebenfalls gerundet ${\displaystyle a_{SOLL}=2\cdot \sin \left({\frac {180^{\circ }}{257}}\right)=2{,}44475829850863E-2\;[LE]}$
• Absoluter Fehler der konstruierten Seite ${\displaystyle F_{a}=a-a_{SOLL}=0\;[LE]}$
• Konstruierter Zentriwinkel in GeoGebra (Anzeige 14 signifikante Nachkommastellen, gerundet) ${\displaystyle \mu =1{,}40077821011673^{\circ }}$
• Zentriwinkel des 257-Ecks, 14 signifikante Nachkommastellen, ebenfalls gerundet ${\displaystyle \mu _{SOLL}={\frac {360^{\circ }}{257}}=1{,}40077821011673^{\circ }}$
• Absoluter Fehler des konstruierten Zentriwinkels ${\displaystyle F_{\mu }=\mu -\mu _{SOLL}=0^{\circ }}$

Beispiel um den Fehler zu verdeutlichen[edit]

Bei einem Radius r = 1 Mrd. km (das Licht bräuchte für diese Strecke ca. 55 min) wäre der absolute Fehler der konstruierten Seitenlänge a < 1 mm.

Approximate construction[edit]

• The method is similar to that of the Tridecagon

Result[edit]

Based on the unit circle r = 1 [unit of length]

• Constructed side of 257-gon in GeoGebra (Display 15 significant decimal places, rounded) ${\displaystyle {\overline {E_{257}E_{1}}}=a=2.44475829850863E-2\;[LE]}$
• Side of the 257-gon, 15 significant decimals, also rounded ${\displaystyle a_{target}=2\cdot \sin \left({\frac {180^{\circ }}{257}}\right)=2.44475829850863E-2\;[LE]}$
• Absolute error of the constructed side in GeoGebra ${\displaystyle F_{a}=a-a_{target}=0\;[unit\;of\;length]}$
• Constructed central angle in GeoGebra (Display 14 significant decimal places, rounded) ${\displaystyle \mu =1.40077821011673^{\circ }}$
• Central angle of the 257-gon in GeoGebra, 14 significant decimals, also rounded ${\displaystyle \mu _{target}={\frac {360^{\circ }}{257}}=1.40077821011673^{\circ }}$
• Absolute error of the constructed central angle ${\displaystyle F_{\mu }=\mu -\mu _{target}=0^{\circ }}$

Example to illustrate the error[edit]

At a radius r = 1 billion km (the light would need about 55 min for this distance) the absolute error of the first side a would be < 1 mm.

Licensing[edit]

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