The rectified dodecagonal duoprism or retwadip is a convex isogonal polychoron that consists of 24 rectified dodecagonal prisms and 144 tetragonal disphenoids . 3 rectified dodecagonal prisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the dodecagonal duoprism .
It can also be formed as the convex hull of 2 oppositely oriented semi-uniform dodecagonal duoprisms , where the edges of one dodecagon are
6
−
2
≈
1.03528
{\displaystyle {\sqrt {6}}-{\sqrt {2}}\approx 1.03528}
Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:
1
+
3
2
{\displaystyle {\frac {1+{\sqrt {3}}}{2}}}
The vertices of a rectified dodecagonal duoprism based on dodecagons of edge length 1, centered at the origin, are given by:
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{\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right),}
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{\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\right),}
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0
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{\displaystyle \left(\pm 2,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}
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{\displaystyle \left(\pm 2,\,0,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(0,\,\pm 2,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}
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{\displaystyle \left(0,\,\pm 2,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(\pm 2,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(0,\,\pm 2,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(\pm 2,\,0,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(0,\,\pm 2,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}
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{\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}
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{\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}
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{\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
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{\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}
