# Rectified pentagonal duoprism

Rectified pentagonal duoprism Rank4
TypeIsogonal
SpaceSpherical
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): $\frac{\sqrt{10}-\sqrt2}{2} ≈ 0.87403$ Edges of pentagons (50): 1
Circumradius$\sqrt{\frac{25-3\sqrt5}{10}} ≈ 1.35247$ 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 $\sqrt5-1 ≈ 1.23607$ times as long as the edges of the other.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{3+\sqrt5}}{2}$ ≈ 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:

• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,\sqrt{\frac{10-2\sqrt5}{5}}\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±1,\,\sqrt{\frac{5-2\sqrt5}{5}}\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{\sqrt5-1}{2},\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0,\,\sqrt{\frac{10-2\sqrt5}{5}}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±1,\,\sqrt{\frac{5-2\sqrt5}{5}}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{\sqrt5-1}{2},\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,\sqrt{\frac{10-2\sqrt5}{5}}\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±1,\,\sqrt{\frac{5-2\sqrt5}{5}}\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{\sqrt5-1}{2},\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(0,\,-\sqrt{\frac{10-2\sqrt5}{5}},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(0,\,-\sqrt{\frac{10-2\sqrt5}{5}},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$ • $\left(0,\,-\sqrt{\frac{10-2\sqrt5}{5}},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$ • $\left(±1,\,-\sqrt{\frac{5-2\sqrt5}{5}},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(±1,\,-\sqrt{\frac{5-2\sqrt5}{5}},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$ • $\left(±1,\,-\sqrt{\frac{5-2\sqrt5}{5}},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$ • $\left(±\frac{\sqrt5-1}{2},\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$ • $\left(±\frac{\sqrt5-1}{2},\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$ • $\left(±\frac{\sqrt5-1}{2},\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right).$ 