# Pentagonal-dodecagonal duoprism

(Redirected from 12-5 duoprism)
Pentagonal-dodecagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Info
Coxeter diagramx5o x12o
SymmetryH2×I2(12), order 240
Elements
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), (2+6)/2 (base 2), and 2 (sides)
Cells12 pentagonal prisms, 5 dodecagonal prisms
Faces60 squares, 12 pentagons, 5 dodecagons,
Edges60+60
Vertices60
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{25+10\sqrt3+\sqrt5}{10}} ≈ 2.11084}$
Hypervolume${\displaystyle \frac{3\sqrt{175+100\sqrt3+70\sqrt5+40\sqrt{15}}}{4} ≈ 19.26273}$
Dichoral anglesTwip–12–twip: 108°
Pip–5–pip: 150°
Twip–4–pip: 90°
Central density1
Euler characteristic0
Number of pieces17
Level of complexity6
Related polytopes
DualPentagonal-dodecagonal duotegum
ConjugatesPentagonal-dodecagrammic duoprism, Pentagrammic-dodecagonal duoprism, Pentagrammic-dodecagrammic duoprism
Properties
ConvexYes
OrientableYes
NatureTame

The pentagonal-dodecagonal duoprism or pitwadip, also known as the 5-12 duoprism, is a uniform duoprism that consists of 5 dodecagonal prisms and 12 pentagonal prisms, with two of each joining at each vertex.

## Vertex coordinates

The coordinates of a pentagonal-dodecagonal duoprism, centered at the origin and with unit edge length, are given by:

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

## Representations

A pentagonal-dodecagonal duoprism has the following Coxeter diagrams:

• x5o x12o (full symmetry)
• x5o x6x (dodecagons as dihexagons)