# Dodecagonal prism

Dodecagonal prism | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Twi |

Coxeter diagram | x2x12o () |

Conway notation | P12 |

Elements | |

Faces | 12 squares, 2 dodecagons |

Edges | 12+24 |

Vertices | 24 |

Vertex figure | Isosceles triangle, edge lengths √2, √2, √2+√3 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 4–4: 150° |

4–12: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 14 |

Level of complexity | 3 |

Related polytopes | |

Army | Twip |

Regiment | Twip |

Dual | Dodecagonal tegum |

Conjugate | Dodecagrammic prism |

Abstract & topological properties | |

Flag count | 144 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Skeleton | GP(12,1) |

Properties | |

Symmetry | I_{2}(12)×A_{1}, order 48 |

Convex | Yes |

Nature | Tame |

The **dodecagonal prism**, or **twip**, is a prismatic uniform polyhedron. It consists of 2 dodecagons and 12 squares. Each vertex joins one dodecagon and two squares. As the name suggests, it is a prism based on a dodecagon.

## Vertex coordinates[edit | edit source]

A dodecagonal prism of edge length 1 has vertex coordinates given by:

- ,
- ,
- .

## Representations[edit | edit source]

A dodecagonal prism has the following Coxeter diagrams:

- x2x12o () (full symmetry)
- x2x6x () (generally a dihexagonal prism)
- s2s12x () (generally a dihexagonal trapezoprism)
- xx12oo&#x (bases separate)
- xx6xx&#x

## Semi-uniform variant[edit | edit source]

The dodecagonal prism has a semi-uniform variant of the form x y12o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by and its volume is given by .

A decagonal prism with base edges of length a and side edges of length b can be alternated to form a hexagonal antiprism with base edges of length and side edges of lengths . In particular if the side edges are times the length of the base edges this gives a uniform pentagonal antiprism.

## Variations[edit | edit source]

A dodecagonal prism has the following variations:

- Dihexagonal prism - prism with dipentagons as bases, and 2 types of rectangles
- Dihexagonal trapezoprism - isogonal with trapezoid sides
- Dodecagonal frustum
- dihexagonal frustum

## External links[edit | edit source]

- Klitzing, Richard. "twip".
- Wikipedia contributors. "Dodecagonal prism".