# Pentagonal-tetrahedral duoprism

Pentagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPetet
Coxeter diagramx5o x3o3o
Elements
Tera5 tetrahedral prisms, 4 triangular-pentagonal duoprisms
Cells5 tetrahedra, 20 triangular prisms, 6 pentagonal prisms
Faces20 triangles, 30 squares, 4 pentagons
Edges20+30
Vertices20
Vertex figureTriangular scalene, edge lengths 1 (base triangle), (1+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {35+4{\sqrt {5}}}{40}}}\approx 1.04814}$
Hypervolume${\displaystyle {\frac {\sqrt {50+20{\sqrt {5}}}}{48}}\approx 0.20276}$
Diteral anglesTepe–tet–tepe: 108°
Tepe–trip–trapedip: 90°
Trapedip–pip–trapedip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsPeg atop trapedip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Pip atop perp pip: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces9
Level of complexity10
Related polytopes
ArmyPetet
RegimentPetet
DualPentagonal-tetrahedral duotegum
ConjugatePentagrammic-tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×H2, order 240
ConvexYes
NatureTame

The pentagonal-tetrahedral duoprism or petet is a convex uniform duoprism that consists of 5 tetrahedral prisms and 4 triangular-pentagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-pentagonal duoprisms.

## Vertex coordinates

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

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

## Representations

A pentagonal-tetrahedral duoprism has the following Coxeter diagrams:

• x5o x3o3o (full symmetry)
• ox3oo xx5oo&#x (pentagon atop triangular-pentagonal duoprism)
• ox xo xx5oo&#x (pentagonal prism atop orthogonal pentagonal prism)