# Pentagonal duoprism

Pentagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymPedip
Coxeter diagramx5o x5o ()
Elements
Cells10 pentagonal prisms
Faces25 squares, 10 pentagons
Edges50
Vertices25
Vertex figureTetragonal disphenoid, edge lengths (1+5)/2 (bases) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5+\sqrt5}{5}} ≈ 1.20300}$
Inradius${\displaystyle \sqrt{\frac{5+2\sqrt5}{20}} ≈ 0.68819}$
Hypervolume${\displaystyle \frac{25+10\sqrt5}{16} ≈ 2.96004}$
Dichoral anglesPip–5–pip: 108°
Pip–4–pip: 90°
Central density1
Number of pieces10
Level of complexity3
Related polytopes
ArmyPedip
RegimentPedip
DualPentagonal duotegum
ConjugatePentagrammic duoprism
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH2≀S2, order 200
ConvexYes
NatureTame

The pentagonal duoprism or pedip, also known as the pentagonal-pentagonal duoprism, the 5 duoprism or the 5-5 duoprism, is a noble uniform duoprism that consists of 10 pentagonal prisms, with 4 meeting at each vertex. It is also the 10-4 gyrochoron and the square funk prism. It is the first in an infinite family of isogonal pentagonal dihedral swirlchora and also the first in an infinite family of isochoric pentagonal hosohedral swirlchora.

A pentagonal duoprism of edge length 1 contains the vertices of a regular pentachoron of edge length ${\displaystyle \sqrt{\frac{5+\sqrt5}{2}}}$, due to the fact the pentachoron is also the 5-2 step prism.

## Vertex coordinates

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

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

## Representations

A pentagonal duoprism has the following Coxeter diagrams:

• x5o x5o (full symmetry)
• ofx xxx5ooo&#xt (pentagonal axial)