# Rectified pentagonal duoprism

Rectified pentagonal duoprism
Rank4
TypeIsogonal
Notation
Bowers style acronymRepdip
Elements
Cells25 tetragonal disphenoids, 10 rectified pentagonal prisms
Faces100 isosceles triangles, 25 squares, 10 pentagons
Edges50+100
Vertices50
Vertex figureWedge
Measures (based on pentagons of edge length 1)
Edge lengthsLacing edges (100): ${\displaystyle {\frac {{\sqrt {10}}-{\sqrt {2}}}{2}}\approx 0.87403}$
Edges of pentagons (50): 1
Circumradius${\displaystyle {\sqrt {\frac {25-3{\sqrt {5}}}{10}}}\approx 1.35247}$
Central density1
Related polytopes
ArmyRepdip
RegimentRepdip
DualJoined pentagonal duotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2≀S2, order 200
ConvexYes
NatureTame

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 ${\displaystyle {\sqrt {5}}-1\approx 1.23607}$ times as long as the edges of the other.

The ratio between the longest and shortest edges is 1:${\displaystyle {\frac {\sqrt {3+{\sqrt {5}}}}{2}}}$ ≈ 1:1.14412.

## Vertex coordinates

The vertices of a rectified pentagonal duoprism with pentagons of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm 1,\,{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm 1,\,{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm 1,\,{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(0,\,-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}},\,0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(0,\,-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(0,\,-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(\pm 1,\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm 1,\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm 1,\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}}\right).}$