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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Trigondodekaeder</span></h1>
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<p>Das <b>Trigondodekaeder</b> (auch <b>Pyramidentetraeder</b>) ist ein <a href="Polyeder" title="Polyeder">Polyeder</a> mit zwölf <a href="Kongruenz_(Geometrie)" title="Kongruenz (Geometrie)">kongruenten</a> <a href="Gleichseitiges_Dreieck" title="Gleichseitiges Dreieck">gleichseitigen Dreiecken</a> als Flächen, 8 Ecken und 18 Kanten. An vier der Ecken grenzen fünf Kanten und an die anderen vier Ecken grenzen vier Kanten an.
</p><p>Es ist ein <a href="Deltaeder" title="Deltaeder">Deltaeder</a> und der <a href="Johnson-K%C3%B6rper" title="Johnson-Körper">Johnson-Körper</a> J<sub>84</sub> von 92, die alle nach dem Mathematiker <a href="Norman_Johnson_(Mathematiker)" title="Norman Johnson (Mathematiker)">Norman Johnson</a> benannt sind.
</p>
<div class="mw-heading mw-heading2"><h2 id="Kartesische_Koordinaten">Kartesische Koordinaten</h2></div>
<p>Die <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a> der Eckpunkte können lauten, bei Mittelpunkt im Ursprung und Kantenlänge 2:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{4}&(&0,&&\;\pm 1,&&\;p&)\\&(&\pm r,&&\;0,&&\;q&)\\&(&0,&&\;\pm r,&&\;-q&)\\&(&\pm 1,&&\;0,&&\;-p&)\end{alignedat}}}">
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<mtd>
<mspace width="thickmathspace"></mspace>
<mi>p</mi>
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<mi></mi>
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<mtd></mtd>
<mtd>
<mi></mi>
<mo stretchy="false">(</mo>
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<mo>±<!-- ± --></mo>
<mi>r</mi>
<mo>,</mo>
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<mtd></mtd>
<mtd>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>,</mo>
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<mtd></mtd>
<mtd>
<mspace width="thickmathspace"></mspace>
<mi>q</mi>
</mtd>
<mtd>
<mi></mi>
<mo stretchy="false">)</mo>
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<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo stretchy="false">(</mo>
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<mtd>
<mn>0</mn>
<mo>,</mo>
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<mtd></mtd>
<mtd>
<mspace width="thickmathspace"></mspace>
<mo>±<!-- ± --></mo>
<mi>r</mi>
<mo>,</mo>
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<mtd></mtd>
<mtd>
<mspace width="thickmathspace"></mspace>
<mo>−<!-- − --></mo>
<mi>q</mi>
</mtd>
<mtd>
<mi></mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo stretchy="false">(</mo>
</mtd>
<mtd>
<mo>±<!-- ± --></mo>
<mn>1</mn>
<mo>,</mo>
</mtd>
<mtd></mtd>
<mtd>
<mspace width="thickmathspace"></mspace>
<mn>0</mn>
<mo>,</mo>
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<mtd></mtd>
<mtd>
<mspace width="thickmathspace"></mspace>
<mo>−<!-- − --></mo>
<mi>p</mi>
</mtd>
<mtd>
<mi></mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}&(&0,&&\;\pm 1,&&\;p&)\\&(&\pm r,&&\;0,&&\;q&)\\&(&0,&&\;\pm r,&&\;-q&)\\&(&\pm 1,&&\;0,&&\;-p&)\end{alignedat}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ce0510e6699904db85e59b0857b011b79f6b8c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:14.064ex; height:12.509ex;" alt="{\displaystyle {\begin{alignedat}{4}&(&0,&&\;\pm 1,&&\;p&)\\&(&\pm r,&&\;0,&&\;q&)\\&(&0,&&\;\pm r,&&\;-q&)\\&(&\pm 1,&&\;0,&&\;-p&)\end{alignedat}}}" loading="lazy"></span></dd></dl>
<p>Nach dem <a href="Satz_von_Pythagoras" class="mw-redirect" title="Satz von Pythagoras">Satz von Pythagoras</a> gilt:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}+(p-q)^{2}=3}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>−<!-- − --></mo>
<mi>q</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>=</mo>
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r^{2}+(p-q)^{2}=3}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f738f0160e8e7f81c6ba651c22713f34329add91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.147ex; height:3.176ex;" alt="{\displaystyle r^{2}+(p-q)^{2}=3}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r-1)^{2}+(p+q)^{2}=4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>+</mo>
<mi>q</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>=</mo>
<mn>4</mn>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle (r-1)^{2}+(p+q)^{2}=4}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2b8e5a320d1a2896d105131fbf67a10b38ca1dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.959ex; height:3.176ex;" alt="{\displaystyle (r-1)^{2}+(p+q)^{2}=4}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}+2q^{2}=2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r^{2}+2q^{2}=2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b65436daba676b8892bea33df5d5cec2c00add22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.5ex; height:3.009ex;" alt="{\displaystyle r^{2}+2q^{2}=2}" loading="lazy"></span></dd></dl>
<p>Daraus folgt:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {1}{2}}({\sqrt {3+2r-r^{2}}}+{\sqrt {3-r^{2}}})\approx 1{,}57}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</msqrt>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo stretchy="false">)</mo>
<mo>≈<!-- ≈ --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>57</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p={\frac {1}{2}}({\sqrt {3+2r-r^{2}}}+{\sqrt {3-r^{2}}})\approx 1{,}57}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64ef7504a4d7517572119641b8c7444a2d002809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:40.149ex; height:5.176ex;" alt="{\displaystyle p={\frac {1}{2}}({\sqrt {3+2r-r^{2}}}+{\sqrt {3-r^{2}}})\approx 1{,}57}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\frac {1}{2}}({\sqrt {3+2r-r^{2}}}-{\sqrt {3-r^{2}}})={\sqrt {\frac {2-r^{2}}{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
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<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
<mo>+</mo>
<mn>2</mn>
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<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mn>2</mn>
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</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q={\frac {1}{2}}({\sqrt {3+2r-r^{2}}}-{\sqrt {3-r^{2}}})={\sqrt {\frac {2-r^{2}}{2}}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8079aae82fe6acb62941573ae8c901ade001ac57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:45.091ex; height:6.176ex;" alt="{\displaystyle q={\frac {1}{2}}({\sqrt {3+2r-r^{2}}}-{\sqrt {3-r^{2}}})={\sqrt {\frac {2-r^{2}}{2}}}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\sqrt {x}}\approx 0{,}411123}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>x</mi>
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<mo>≈<!-- ≈ --></mo>
<mn>0,411</mn>
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<annotation encoding="application/x-tex">{\displaystyle q={\sqrt {x}}\approx 0{,}411123}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6ad8d0c4d2b43e6f657f23199298cfc323c8595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.316ex; height:3.009ex;" alt="{\displaystyle q={\sqrt {x}}\approx 0{,}411123}" loading="lazy"></span> als eine von drei Lösungen der Gleichung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{3}+11x^{2}+4x-1=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msup>
<mo>+</mo>
<mn>11</mn>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>+</mo>
<mn>4</mn>
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<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2x^{3}+11x^{2}+4x-1=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e70f41dbf4e0d6b100e4df46f90a262e52e78030.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.692ex; height:2.843ex;" alt="{\displaystyle 2x^{3}+11x^{2}+4x-1=0}" loading="lazy"></span>.</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\approx 1{,}289169}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>≈<!-- ≈ --></mo>
<mn>1,289</mn>
<mn>169</mn>
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<annotation encoding="application/x-tex">{\displaystyle r\approx 1{,}289169}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a32779933710f66472ff9fdd1c6b96b6c15541e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.931ex; height:2.509ex;" alt="{\displaystyle r\approx 1{,}289169}" loading="lazy"></span> als eine von drei Lösungen der Gleichung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{3}-3r^{2}-4r+8=0}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msup>
<mo>−<!-- − --></mo>
<mn>3</mn>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mi>r</mi>
<mo>+</mo>
<mn>8</mn>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r^{3}-3r^{2}-4r+8=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49aa105ab7b35d4698f0410409b7977ed2df2a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.524ex; height:2.843ex;" alt="{\displaystyle r^{3}-3r^{2}-4r+8=0}" loading="lazy"></span>.</dd></dl>
<p>Aus den Koordinaten ergibt sich, dass ein minimaler Quader, der den Körper einschließt, die Form einer <a href="Quader" title="Quader">Quadratischen Säule</a> mit einer Höhe von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff0d3e38652941fc7a2844611f6c7fa8a23104a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.332ex; height:2.509ex;" alt="{\displaystyle 2p}" loading="lazy"></span> und einer Breite von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r{\sqrt {2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r{\sqrt {2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c8ffb58fbf20d3cac6f39bfd26d9ffa3b67f35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.147ex; height:3.009ex;" alt="{\displaystyle r{\sqrt {2}}}" loading="lazy"></span> annimmt. Alle sechs Flächen der Säule berühren eine Kante des Trigondodekaeders. Bedingt durch die Tatsache, dass der zuvor genannte <a href="H%C3%BCllk%C3%B6rper" title="Hüllkörper">Hüllkörper</a> kein <a href="W%C3%BCrfel_(Geometrie)" title="Würfel (Geometrie)">Würfel</a> ist, besitzt das Trigondodekaeder weder eine <a href="Umkugel" title="Umkugel">Umkugel</a> noch eine <a href="Inkugel" title="Inkugel">Inkugel</a> oder <a href="Kantenkugel" title="Kantenkugel">Kantenkugel</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Formeln">Formeln</h2></div>
<table class="wikitable">
<tbody><tr>
<th colspan="2" style="background:#C0C0FF">Größen eines Trigondodekaeders mit Kantenlänge <i>a</i>
</th></tr>
<tr>
<td class="hintergrundfarbe5"><b><a href="Volumen" title="Volumen">Volumen</a></b>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {r}{12}}a^{3}\left(6{\sqrt {3-r^{2}}}+r{\sqrt {4-2r^{2}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>r</mi>
<mn>12</mn>
</mfrac>
</mrow>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>+</mo>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>4</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V={\frac {r}{12}}a^{3}\left(6{\sqrt {3-r^{2}}}+r{\sqrt {4-2r^{2}}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31dc8cfafd130dbd1ccfbe0b577dfbb787c41a8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.566ex; height:4.843ex;" alt="{\displaystyle V={\frac {r}{12}}a^{3}\left(6{\sqrt {3-r^{2}}}+r{\sqrt {4-2r^{2}}}\right)}" loading="lazy"></span>
</td></tr>
<tr>
<td class="hintergrundfarbe5"><b><a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Oberflächeninhalt</a></b>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{O}=3a^{2}{\sqrt {3}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>O</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>3</mn>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{O}=3a^{2}{\sqrt {3}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1551cda07eea4d9f7a4c5dfc8316de34e82639b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.873ex; height:3.009ex;" alt="{\displaystyle A_{O}=3a^{2}{\sqrt {3}}}" loading="lazy"></span>
</td></tr>
<tr>
<td class="hintergrundfarbe5"><b>1. <a href="Fl%C3%A4chenwinkel" class="mw-redirect" title="Flächenwinkel">Flächenwinkel</a><br> ≈ 96,2°</b>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \,\alpha _{1}={\frac {1}{3}}\left(3-2r^{2}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos \,\alpha _{1}={\frac {1}{3}}\left(3-2r^{2}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4051b5e74aa089fdea545ebaed861352da8782fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.309ex; height:5.176ex;" alt="{\displaystyle \cos \,\alpha _{1}={\frac {1}{3}}\left(3-2r^{2}\right)}" loading="lazy"></span>
</td></tr>
<tr>
<td class="hintergrundfarbe5"><b>2. Flächenwinkel<br> ≈ 121,74°</b>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \,\alpha _{2}={\frac {1}{3}}\left(1-2r\right)={\frac {1}{3}}\left(4q^{2}-1\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos \,\alpha _{2}={\frac {1}{3}}\left(1-2r\right)={\frac {1}{3}}\left(4q^{2}-1\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/707caf2ac2d07e2a109031145eff16625ecf9030.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.848ex; height:5.176ex;" alt="{\displaystyle \cos \,\alpha _{2}={\frac {1}{3}}\left(1-2r\right)={\frac {1}{3}}\left(4q^{2}-1\right)}" loading="lazy"></span>
</td></tr>
<tr>
<td class="hintergrundfarbe5"><b>3. Flächenwinkel<br> ≈ 166,44°</b>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \,\alpha _{3}={\frac {2}{3}}\left(1-p^{2}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mspace width="thinmathspace"></mspace>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos \,\alpha _{3}={\frac {2}{3}}\left(1-p^{2}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab6993828424ce71279be4b83f176ea9eff023b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.268ex; height:5.176ex;" alt="{\displaystyle \cos \,\alpha _{3}={\frac {2}{3}}\left(1-p^{2}\right)}" loading="lazy"></span>
</td></tr>
<tr>
<td class="hintergrundfarbe5"><b> <a href="Sph%C3%A4rizit%C3%A4t_(Geologie)" title="Sphärizität (Geologie)">Sphärizität</a><br> ≈ 0,84133</b>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi ={\frac {\sqrt[{3}]{36\,\pi \,V^{2}}}{A_{O}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mroot>
<mrow>
<mn>36</mn>
<mspace width="thinmathspace"></mspace>
<mi>π<!-- π --></mi>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</mroot>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>O</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Psi ={\frac {\sqrt[{3}]{36\,\pi \,V^{2}}}{A_{O}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ec251eed3703de8e8398a639d1b20c85619aed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.081ex; height:6.343ex;" alt="{\displaystyle \Psi ={\frac {\sqrt[{3}]{36\,\pi \,V^{2}}}{A_{O}}}}" loading="lazy"></span>
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Triakistetraeder" title="Triakistetraeder">Triakistetraeder</a>, das von zwölf gleichschenkligen Dreiecken begrenzt wird.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Commons"></span></span></div><b><span class=""><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Snub_disphenoid?uselang=de"><span lang="en">Commons</span>: Trigondodekaeder</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</div>
<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TrigonalDodecahedron.html"><i>Trigondodekaeder</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/JohnsonSolid.html"><i>Die Johnson-Körper</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-04-27" href="https://de.wikipedia.org/wiki/?title=Trigondodekaeder&oldid=244458984">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
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