# Triangular-pentagonal duoprism

(Redirected from 5-3 duoprism)
Triangular-pentagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Bowers style acronymTrapedip
Info
Coxeter diagramx3o x5o
SymmetryA2×H2, order 60
ArmyTrapedip
RegimentTrapedip
Elements
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), (1+5)/2 (base 2), and 2 (sides)
Cells5 triangular prisms, 3 pentagonal prisms
Faces5 triangles, 15 squares, 3 pentagons
Edges15+15
Vertices15
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{25+3\sqrt5}{30}} ≈ 1.02808}$
Hypervolume${\displaystyle \frac{\sqrt{75+30\sqrt5}}{16} ≈ 0.74499}$
Dichoral anglesTrip–3–trip: 108°
Trip–4–pip: 90°
Pip–5–pip: 60°
Height${\displaystyle \frac{\sqrt3}2 ≈ 0.86603}$
Central density1
Euler characteristic0
Number of pieces8
Level of complexity6
Related polytopes
DualTriangular-pentagonal duotegum
ConjugateTriangular-pentagrammic duoprism
Properties
ConvexYes
OrientableYes
NatureTame

The triangular-pentagonal duoprism or trapedip, also known as the 3-5 duoprism, is a uniform duoprism that consists of 3 pentagonal prisms and 5 triangular prisms, with two of each at each vertex.

It is also a CRF segmentochoron, as pentagon atop pentagonal prism. It is designated K-4.34 on Richard Klitzing's list.

## Vertex coordinates

Coordinates for the vertices of a triangular-pentagonal duoprism with edge length 1 are given by:

• ${\displaystyle \left(0,\,\frac{\sqrt3}{3},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt3}{3},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),}$

## Representations

A triangular-pentagonal duoprism has the following Coxeter diagrams:

• x3o x5o (full symmetry)
• ox xx5oo&#x (pentagon atop pentagon prism)
• ofx xxx3oooo&#xt (triangle-first)