The rectified pentagonal duoprism or repdip is a convex isogonal polychoron that consists of 10 rectified pentagonal prisms and 25 tetragonal disphenoids . 3 rectified pentagonal prisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the pentagonal duoprism .
It can also be formed as the convex hull of 2 oppositely oriented semi-uniform pentagonal duoprisms , where the edges of one pentagon are
5
−
1
≈
1.23607
{\displaystyle {\sqrt {5}}-1\approx 1.23607}
times as long as the edges of the other.
The ratio between the longest and shortest edges is 1:
3
+
5
2
{\displaystyle {\frac {\sqrt {3+{\sqrt {5}}}}{2}}}
≈ 1:1.14412.
The vertices of a rectified pentagonal duoprism with pentagons of edge length 1, centered at the origin, are given by:
(
0
,
5
+
5
10
,
0
,
10
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2
5
5
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\right),}
(
0
,
5
+
5
10
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±
1
,
5
−
2
5
5
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm 1,\,{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right),}
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,
5
+
5
10
,
±
5
−
1
2
,
−
5
+
5
10
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
(
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1
+
5
4
,
5
−
5
40
,
0
,
10
−
2
5
5
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\right),}
(
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1
+
5
4
,
5
−
5
40
,
±
1
,
5
−
2
5
5
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm 1,\,{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right),}
(
±
1
+
5
4
,
5
−
5
40
,
±
5
−
1
2
,
−
5
+
5
10
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
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1
2
,
−
5
+
2
5
20
,
0
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10
−
2
5
5
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\right),}
(
±
1
2
,
−
5
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2
5
20
,
±
1
,
5
−
2
5
5
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm 1,\,{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right),}
(
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1
2
,
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5
+
2
5
20
,
±
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−
1
2
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−
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+
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10
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
(
0
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−
10
−
2
5
5
,
0
,
−
5
+
5
10
)
,
{\displaystyle \left(0,\,-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}},\,0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
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−
10
−
2
5
5
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±
1
+
5
4
,
−
5
−
5
40
)
,
{\displaystyle \left(0,\,-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}
(
0
,
−
10
−
2
5
5
,
±
1
2
,
5
+
2
5
20
)
,
{\displaystyle \left(0,\,-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}
(
±
1
,
−
5
−
2
5
5
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0
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−
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+
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10
)
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{\displaystyle \left(\pm 1,\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
(
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1
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−
5
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2
5
5
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±
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+
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4
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−
5
40
)
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{\displaystyle \left(\pm 1,\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}
(
±
1
,
−
5
−
2
5
5
,
±
1
2
,
5
+
2
5
20
)
,
{\displaystyle \left(\pm 1,\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}
(
±
5
−
1
2
,
5
+
5
10
,
0
,
−
5
+
5
10
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}
(
±
5
−
1
2
,
5
+
5
10
,
±
1
+
5
4
,
−
5
−
5
40
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}
(
±
5
−
1
2
,
5
+
5
10
,
±
1
2
,
5
+
2
5
20
)
.
{\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right).}