# Pentagonal-octahedral duoprism

Pentagonal-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPoct
Coxeter diagramx5o o4o3x
Elements
Tera5 octahedral prisms, 8 triangular-pentagonal duoprisms
Cells40 triangular prisms, 5 octahedra, 12 pentagonal prisms
Faces40 triangles, 60 squares, 6 pentagons
Edges30+60
Vertices30
Vertex figureSquare scalene, edge lengths 1 (base square), (1+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {10+{\sqrt {5}}}{10}}}\approx 1.10617}$
Hypervolume${\displaystyle {\frac {\sqrt {50+20{\sqrt {5}}}}{12}}\approx 0.81104}$
Diteral anglesTrapedip–pip–trapedip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Ope–oct–ope: 108°
Trapedip–trip–ope: 90°
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces13
Level of complexity10
Related polytopes
ArmyPoct
RegimentPoct
DualPentagonal-cubic duotegum
ConjugatePentagrammic-octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×H2, order 480
ConvexYes
NatureTame

The pentagonal-octahedral duoprism or poct is a convex uniform duoprism that consists of 5 octahedral prisms and 8 triangular-pentagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-pentagonal duoprisms.

## Vertex coordinates

The vertices of a pentagonal-octahedral duoprism of edge length 1 are given by all permutations and sign changes of the last three coordinates of:

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

## Representations

A pentagonal-octahedral duoprism has the following Coxeter diagrams:

• x5o o4o3x (full symmetry)
• x5o o3x3o (octahedra as tetratetrahedra)
• xo3ox xx5oo&#x (triangular-pentagonal duoprism atop triangle-gyrated triangular-pentagonal duoprism)