File:01 Würfelhalbierung-Zissoide v. Diokles.svg

Summary[edit]

Description01 Würfelhalbierung-Zissoide v. Diokles.svg	Deutsch: Würfelhalbierung, Konstruktion der Kantenlänge ${\displaystyle a\cdot {\sqrt[{3}]{2}}}$ eines Würfels, der exakt das halbe Volumen des Ausgangswürfel hat, mithilfe der Zissoide v. Diokles. English: Halving the cube, construction of edge length ${\displaystyle a\cdot {\sqrt[{3}]{2}}}$ of a cube which has exactly half the volume of the given cube, with the help of the cissoid v. Diocles.
Date	16 November 2021
Source	Own work
Author	Petrus3743
SVG development InfoField	The SVG code is valid.   This trigonometry was created with GeoGebra by Petrus3743.

Nachrechnung[edit]

Es sei

${\displaystyle {\overline {MA}}=a}$,

dann ist

${\displaystyle {\overline {ME}}={\overline {MF}}=a\cdot {\sqrt[{3}]{2}}}$

die Kantenlänge des verdoppelten Würfels. Da ${\displaystyle \Delta \;FMB}$ und ${\displaystyle \Delta \;PMG}$ ähnliche Dreiecke sind, gilt

${\displaystyle {\overline {MB}}:{\overline {MF}}={\overline {MG}}:{\overline {MP}}.}$

Mit den eingesetzten Werten bestätigt sich:

${\displaystyle {\frac {a}{a\cdot {\sqrt[{3}]{2}}}}={\frac {\overline {MG}}{a}}\;\Rightarrow }$

${\displaystyle {\overline {MG}}={\frac {1}{\sqrt[{3}]{2}}}.}$

Vergleich der Volumina

Mit der Kantenlänge ${\displaystyle a=1}$ des Ausgangswürfels ${\displaystyle W_{1}}$ gilt

${\displaystyle V_{W_{1}}=1^{3}\;[VE],}$

mit der Kantenlänge ${\displaystyle {\overline {MG}}={\frac {1}{\sqrt[{3}]{2}}}}$ des konstruierten Würfels ${\displaystyle W_{2}}$ gilt

${\displaystyle V_{W_{2}}=\left({\frac {1}{\sqrt[{3}]{2}}}\right)^{3}={\frac {1}{2}}\;[VE].}$

Somit kann der Würfel ${\displaystyle W_{2}}$ mithilfe der konstruierten Würfelkantenlänge ${\displaystyle a\cdot {\frac {1}{\sqrt[{3}]{2}}}}$, als exakte Halbierung des Ausgangswürfels ${\displaystyle W_{1}}$ dargestellt werden.

Licensing[edit]

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