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 \sqrt5-1 ≈ 1.23607}
The ratio between the longest and shortest edges is 1:
3
+
5
2
{\displaystyle \frac{\sqrt{3+\sqrt5}}{2}}
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
−
2
5
5
)
,
{\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,\sqrt{\frac{10-2\sqrt5}{5}}\right),}
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0
,
5
+
5
10
,
±
1
,
5
−
2
5
5
)
,
{\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±1,\,\sqrt{\frac{5-2\sqrt5}{5}}\right),}
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0
,
5
+
5
10
,
±
5
−
1
2
,
−
5
+
5
10
)
,
{\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{\sqrt5-1}{2},\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}
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±
1
+
5
4
,
5
−
5
40
,
0
,
10
−
2
5
5
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0,\,\sqrt{\frac{10-2\sqrt5}{5}}\right),}
(
±
1
+
5
4
,
5
−
5
40
,
±
1
,
5
−
2
5
5
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±1,\,\sqrt{\frac{5-2\sqrt5}{5}}\right),}
(
±
1
+
5
4
,
5
−
5
40
,
±
5
−
1
2
,
−
5
+
5
10
)
,
{\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{\sqrt5-1}{2},\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}
(
±
1
2
,
−
5
+
2
5
20
,
0
,
10
−
2
5
5
)
,
{\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,\sqrt{\frac{10-2\sqrt5}{5}}\right),}
(
±
1
2
,
−
5
+
2
5
20
,
±
1
,
5
−
2
5
5
)
,
{\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±1,\,\sqrt{\frac{5-2\sqrt5}{5}}\right),}
(
±
1
2
,
−
5
+
2
5
20
,
±
5
−
1
2
,
−
5
+
5
10
)
,
{\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{\sqrt5-1}{2},\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}
(
0
,
−
10
−
2
5
5
,
0
,
−
5
+
5
10
)
,
{\displaystyle \left(0,\,-\sqrt{\frac{10-2\sqrt5}{5}},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}
(
0
,
−
10
−
2
5
5
,
±
1
+
5
4
,
−
5
−
5
40
)
,
{\displaystyle \left(0,\,-\sqrt{\frac{10-2\sqrt5}{5}},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),}
(
0
,
−
10
−
2
5
5
,
±
1
2
,
5
+
2
5
20
)
,
{\displaystyle \left(0,\,-\sqrt{\frac{10-2\sqrt5}{5}},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),}
(
±
1
,
−
5
−
2
5
5
,
0
,
−
5
+
5
10
)
,
{\displaystyle \left(±1,\,-\sqrt{\frac{5-2\sqrt5}{5}},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}
(
±
1
,
−
5
−
2
5
5
,
±
1
+
5
4
,
−
5
−
5
40
)
,
{\displaystyle \left(±1,\,-\sqrt{\frac{5-2\sqrt5}{5}},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),}
(
±
1
,
−
5
−
2
5
5
,
±
1
2
,
5
+
2
5
20
)
,
{\displaystyle \left(±1,\,-\sqrt{\frac{5-2\sqrt5}{5}},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),}
(
±
5
−
1
2
,
5
+
5
10
,
0
,
−
5
+
5
10
)
,
{\displaystyle \left(±\frac{\sqrt5-1}{2},\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),}
(
±
5
−
1
2
,
5
+
5
10
,
±
1
+
5
4
,
−
5
−
5
40
)
,
{\displaystyle \left(±\frac{\sqrt5-1}{2},\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),}
(
±
5
−
1
2
,
5
+
5
10
,
±
1
2
,
5
+
2
5
20
)
.
{\displaystyle \left(±\frac{\sqrt5-1}{2},\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right).}
