# Pentagonal antiprism

Pentagonal antiprism | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Pap |

Coxeter diagram | s2s10o () |

Conway notation | A5 |

Elements | |

Faces | 10 triangles, 2 pentagons |

Edges | 10+10 |

Vertices | 10 |

Vertex figure | Isosceles trapezoid, edge lengths 1, 1, 1, (1+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–3: |

5–3: | |

Height | |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 4 |

Related polytopes | |

Army | Pap |

Regiment | Pap |

Dual | Pentagonal antitegum |

Conjugate | Pentagrammic retroprism |

Abstract & topological properties | |

Flag count | 80 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | (I_{2}(10)×A_{1})/2, order 20 |

Convex | Yes |

Nature | Tame |

The **pentagonal antiprism**, or **pap**, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagons. Each vertex joins one pentagon and three triangles. As the name suggests, it is an antiprism based on a pentagon.

It can also be obtained as a diminishing of the regular icosahedron when two pentagonal pyramids are removed from opposite ends.

## Vertex coordinates[edit | edit source]

A pentagonal antiprism of edge length 1 has vertex coordinates given by:

These coordinates are obtained by removing two opposite vertices from a regular icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points:

## Representations[edit | edit source]

A pentagonal antiprism has the following Coxeter diagrams:

- s2s10o (alternated decagonal prism)
- s2s5s (alternated dipentagonal prism)
- xo5ox&#x (bases considered separately)

## General variant[edit | edit source]

The pentagonal antiprism has a general isogonal variant of the form xo5ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by .

The bases of the pentagonal antiprism are rotated from each other by an angle of 36°. If this angle is changed the result is more properly called a pentagonal gyroprism.

A notable case occurs as the alternation of the uniform decagonal prism. This specific case has base edges of length and side edges of length .

## Related polyhedra[edit | edit source]

A pentagonal pyramid can be attached to a base of the pentagonal antiprism to form the gyroelongated pentagonal pyramid. If a second pyramid is attached to the other base, the result is the gyroelongated pentagonal bipyramid, better known as the regular icosahedron.

Two non-prismatic uniform polyhedron compounds are composed of pentagonal antiprisms:

There are also an infinite amount of prismatic uniform compounds that are the antiprisms of compounds of pentagons.

## External links[edit | edit source]

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#3 under ike).

- Klitzing, Richard. "pap".

- Quickfur. "The Pentagonal Antiprism".

- Wikipedia Contributors. "Pentagonal antiprism".
- McCooey, David. "Pentagonal Antiprism"