# Rectified dodecagonal duoprism

Rectified dodecagonal duoprism
Rank4
TypeIsogonal
Notation
Elements
Cells144 tetragonal disphenoids, 24 rectified dodecagonal prisms
Faces576 isosceles triangles, 144 squares, 24 dodecagons
Edges288+576
Vertices288
Vertex figureWedge
Measures (based on dodecagons of edge length 1)
Edge lengthsLacing edges (576): ${\displaystyle {\sqrt {3}}-1\approx 0.73205}$
Edges of dodecagons (288): 1
Circumradius${\displaystyle {\sqrt {6+{\sqrt {3}}}}\approx 2.78066}$
Central density1
Related polytopes
DualJoined dodecagonal duotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12)≀S2, order 1152
ConvexYes
NatureTame

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 ${\displaystyle {\sqrt {6}}-{\sqrt {2}}\approx 1.03528}$ times as long as the edges of the other.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\displaystyle {\frac {1+{\sqrt {3}}}{2}}}$ ≈ 1:1.36603.

## Vertex coordinates

The vertices of a rectified dodecagonal duoprism based on dodecagons of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}},\,\pm {\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\right),}$
• ${\displaystyle \left(\pm 2,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}$
• ${\displaystyle \left(\pm 2,\,0,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm 2,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}$
• ${\displaystyle \left(0,\,\pm 2,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(\pm 2,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm 2,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(\pm 2,\,0,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm 2,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}$
• ${\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{,}}\,0\right),}$
• ${\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {3}},\,\pm 1,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$
• ${\displaystyle \left(\pm 1,\,\pm {\sqrt {3}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {6}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\right),}$