# Rectified decagonal duoprism

Rectified decagonal duoprism
Rank4
TypeIsogonal
SpaceSpherical
Notation
Bowers style acronymRededip
Elements
Cells100 tetragonal disphenoids, 20 rectified decagonal prisms
Faces400 isosceles triangles, 100 squares, 20 decagons
Edges200+400
Vertices200
Vertex figureWedge
Measures (based on decagons of edge length 1)
Edge lengthsLacing edges (400): ${\displaystyle \sqrt{\frac{5-\sqrt5}{5}} ≈ 0.74350}$
Edges of decagons (200): 1
Circumradius${\displaystyle \sqrt{\frac{35+9\sqrt5}{20}} ≈ 2.34786}$
Central density1
Related polytopes
ArmyRededip
RegimentRededip
DualJoined decagonal duotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(10)≀S2, order 800
ConvexYes
NatureTame

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 ${\displaystyle \sqrt{\frac{10-2\sqrt5}{5}} ≈ 1.05146}$ times as long as the edges of the other.

The ratio between the longest and shortest edges is 1:${\displaystyle \frac{\sqrt{5+\sqrt5}}{2}}$ ≈ 1:1.34500.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,0,\,±\sqrt{\frac{10+2\sqrt5}{5}}\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±1,\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5-\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,0,\,±\sqrt{\frac{10+2\sqrt5}{5}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±1,\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5-\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,0,\,±\sqrt{\frac{10+2\sqrt5}{5}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±1,\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,±\frac{1+\sqrt5}{2},\,±\sqrt{\frac{5-\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{10+2\sqrt5}{5}},\,0,\,±\frac{1+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(±\sqrt{\frac{10+2\sqrt5}{5}},\,0,\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{10+2\sqrt5}{5}},\,0,\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±1,\,±\frac{1+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±1,\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±1,\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5-\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5-\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{5-\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right).}$