# Decagonal prism

Decagonal prism | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Dip |

Coxeter diagram | x x10o () |

Conway notation | P10 |

Elements | |

Faces | 10 squares, 2 decagons |

Edges | 10+20 |

Vertices | 20 |

Vertex figure | Isosceles triangle, edge lengths √2, √2, √(5+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 4–4: 144° |

4–10: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 3 |

Related polytopes | |

Army | Dip |

Regiment | Dip |

Dual | Decagonal tegum |

Conjugate | Decagrammic prism |

Abstract & topological properties | |

Flag count | 120 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Skeleton | GP(10,1) |

Properties | |

Symmetry | I_{2}(10)×A_{1}, order 40 |

Convex | Yes |

Nature | Tame |

The **decagonal prism**, or **dip**, is a prismatic uniform polyhedron. It consists of 2 decagons and 10 squares. Each vertex joins one decagon and two squares. As the name suggests, it is a prism based on a decagon.

It is the highest convex polygonal prism to occur as cells in uniform polychora.

## Vertex coordinates[edit | edit source]

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

- ,
- ,
- .

## Representations[edit | edit source]

A decagonal prism has the following Coxeter diagrams:

- x2x10o () (full symmetry)
- x2x5x () (as dipentagonal prism)
- s2s10x () (as dipentagonal trapezoprism)
- xx10oo&#x (decagonal frustum)
- xx5xx&#x (dipentagonal frustum)
- xxxxx xFVFx&#xt (A1×A1 axial, square-first)

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

The decagonal prism has a semi-uniform variant of the form x y10o 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 pentagonal 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 decagonal prism has the following variations:

- Dipentagonal prism - prism with dipentagons as bases, and 2 types of rectangles
- Dipentagonal trapezoprism - isogonal with trapezoid sides
- Decagonal frustum
- Dipentagonal frustum

## Related polyhedra[edit | edit source]

A number of Johnson solids can be formed by attaching various configurations of pentagonal cupolas and pentagonal rotundas to the bases of the decagonal prism:

- Elongated pentagonal cupola - Cupola attached to one base
- Elongated pentagonal rotunda - Rotunda attached to one base
- Elongated pentagonal orthobicupola - Cupolas in same orientation attached to both bases
- Elongated pentagonal gyrobicupola - Cupolas rotated by 36º attached to bases
- Elongated pentagonal orthocupolarotunda - Cupola attached to one base, rotunda with same pentagon orientation attached to other base
- Elongated pentagonal gyrocupolarotunda - Cupola attached to one base, rotunda with pentagon rotated by 36º attached to other base
- Elongated pentagonal orthobirotunda - Rotundas in same orientation attached to both bases
- Elongated pentagonal gyrobirotunda - Rotundas rotated by 36º attached to bases

The rhombisnub dodecahedron is a uniform polyhedron compound composed of 6 decagonal prisms.

## External links[edit | edit source]

- Klitzing, Richard. "dip".
- Quickfur. "The Decagonal Prism".

- Wikipedia contributors. "Decagonal prism".
- Hi.gher.Space Wiki Contributors. "Decagonal prism".