The rectified octagonal duoprism or reodip is a convex isogonal polychoron that consists of 16 rectified octagonal prisms and 64 tetragonal disphenoids . 3 rectified octagonal prisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the octagonal duoprism .
It can also be formed as the convex hull of 2 oppositely oriented semi-uniform octagonal duoprisms , where the edges of one octagon are
4
−
2
2
≈
1.08239
{\displaystyle {\sqrt {4-2{\sqrt {2}}}}\approx 1.08239}
The ratio between the longest and shortest edges is 1:
2
+
2
2
{\displaystyle {\sqrt {\frac {2+{\sqrt {2}}}{2}}}}
The vertices of a rectified octagonal duoprism based on octagons of edge length 1, centered at the origin, are given by:
(
±
1
2
,
±
1
+
2
2
,
±
2
−
2
2
,
±
2
+
2
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\sqrt {\frac {2-{\sqrt {2}}}{2}}},\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}}\right),}
(
±
1
2
,
±
1
+
2
2
,
±
2
+
2
2
,
±
2
−
2
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,\pm {\sqrt {\frac {2-{\sqrt {2}}}{2}}}\right),}
(
±
1
+
2
2
,
±
1
2
,
±
2
−
2
2
,
±
2
+
2
2
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {\frac {2-{\sqrt {2}}}{2}}},\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}}\right),}
(
±
1
+
2
2
,
±
1
2
,
±
2
+
2
2
,
±
2
−
2
2
)
.
{\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,\pm {\sqrt {\frac {2-{\sqrt {2}}}{2}}}\right).}
(
±
2
,
0
,
±
2
+
2
2
,
0
)
,
{\displaystyle \left(\pm {\sqrt {2}},\,0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,0\right),}
(
±
2
,
0
,
0
,
±
2
+
2
2
)
,
{\displaystyle \left(\pm {\sqrt {2}},\,0,\,0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}}\right),}
(
0
,
±
2
,
±
2
+
2
2
,
0
)
,
{\displaystyle \left(0,\,\pm {\sqrt {2}},\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,0\right),}
(
0
,
±
2
,
0
,
±
2
+
2
2
)
,
{\displaystyle \left(0,\,\pm {\sqrt {2}},\,0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}}\right),}
(
±
1
,
±
1
,
±
2
+
2
2
,
0
)
,
{\displaystyle \left(\pm 1,\,\pm 1,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,0\right),}
(
±
1
,
±
1
,
0
,
±
2
+
2
2
)
,
{\displaystyle \left(\pm 1,\,\pm 1,\,0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}}\right),}
(
±
2
,
0
,
±
2
+
2
2
,
±
2
+
2
2
)
,
{\displaystyle \left(\pm {\sqrt {2}},\,0,\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\right),}
(
0
,
±
2
,
±
2
+
2
2
,
±
2
+
2
2
)
,
{\displaystyle \left(0,\,\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\right),}
(
±
1
,
±
1
,
±
2
+
2
2
,
±
2
+
2
2
)
,
{\displaystyle \left(\pm 1,\,\pm 1,\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\right),}
