The rectified decagonal duoprism or rededip is a convex isogonal polychoron that consists of 20 rectified decagonal prisms and 100 tetragonal disphenoids . 3 rectified decagonal duoprisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the decagonal duoprism .
It can also be formed as the convex hull of 2 oppositely oriented semi-uniform decagonal duoprisms , where the edges of one decagon are
10
−
2
5
5
≈
1.05146
{\displaystyle \sqrt{\frac{10-2\sqrt5}{5}} ≈ 1.05146}
The ratio between the longest and shortest edges is 1:
5
+
5
2
{\displaystyle \frac{\sqrt{5+\sqrt5}}{2}}
The vertices of a rectified decagonal duoprism based on decagons of edge length 1, centered at the origin, are given by:
(
0
,
±
1
+
5
2
,
0
,
±
10
+
2
5
5
)
,
{\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,0,\,±\sqrt{\frac{10+2\sqrt5}{5}}\right),}
(
0
,
±
1
+
5
2
,
±
1
,
±
5
+
2
5
5
)
,
{\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±1,\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),}
(
0
,
±
1
+
5
2
,
±
1
+
5
2
,
±
5
−
5
10
)
,
{\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5-\sqrt5}{10}}\right),}
(
±
5
+
5
8
,
±
3
+
5
4
,
0
,
±
10
+
2
5
5
)
,
{\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,0,\,±\sqrt{\frac{10+2\sqrt5}{5}}\right),}
(
±
5
+
5
8
,
±
3
+
5
4
,
±
1
,
±
5
+
2
5
5
)
,
{\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±1,\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),}
(
±
5
+
5
8
,
±
3
+
5
4
,
±
1
+
5
2
,
±
5
−
5
10
)
,
{\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5-\sqrt5}{10}}\right),}
(
±
5
+
2
5
2
,
±
1
2
,
0
,
±
10
+
2
5
5
)
,
{\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,0,\,±\sqrt{\frac{10+2\sqrt5}{5}}\right),}
(
±
5
+
2
5
2
,
±
1
2
,
±
1
,
±
5
+
2
5
5
)
,
{\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±1,\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),}
(
±
5
+
2
5
2
,
±
1
2
,
±
1
+
5
2
,
±
5
−
5
10
)
,
{\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5-\sqrt5}{10}}\right),}
(
±
10
+
2
5
5
,
0
,
±
1
+
5
2
,
0
)
,
{\displaystyle \left(±\sqrt{\frac{10+2\sqrt5}{5}},\,0,\,±\frac{1+\sqrt5}{2},\,0\right),}
(
±
10
+
2
5
5
,
0
,
±
3
+
5
4
,
±
5
+
5
8
)
,
{\displaystyle \left(±\sqrt{\frac{10+2\sqrt5}{5}},\,0,\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}
(
±
10
+
2
5
5
,
0
,
±
1
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\sqrt{\frac{10+2\sqrt5}{5}},\,0,\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}
(
±
5
+
2
5
5
,
±
1
,
±
1
+
5
2
,
0
)
,
{\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±1,\,±\frac{1+\sqrt5}{2},\,0\right),}
(
±
5
+
2
5
5
,
±
1
,
±
3
+
5
4
,
±
5
+
5
8
)
,
{\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±1,\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}
(
±
5
+
2
5
5
,
±
1
,
±
1
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±1,\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}
(
±
5
−
5
10
,
±
1
+
5
2
,
±
1
+
5
2
,
0
)
,
{\displaystyle \left(±\sqrt{\frac{5-\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,0\right),}
(
±
5
−
5
10
,
±
1
+
5
2
,
±
3
+
5
4
,
±
5
+
5
8
)
,
{\displaystyle \left(±\sqrt{\frac{5-\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}
(
±
5
−
5
10
,
±
1
+
5
2
,
±
1
2
,
±
5
+
2
5
2
)
.
{\displaystyle \left(±\sqrt{\frac{5-\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right).}
