# Pentagonal tegum

Pentagonal tegum | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Pet |

Coxeter diagram | oxo5ooo&#xt |

Elements | |

Faces | 10 triangles |

Edges | 5+10 |

Vertices | 2+5 |

Vertex figures | 2 pentagons, edge length 1 |

5 rhombi, edge length 1 | |

Measures (edge length 1) | |

Inradius | |

Volume | |

Dihedral angles | 3-3 pyramidal: |

3-3 equatorial: | |

Height | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 3 |

Related polytopes | |

Army | Pet |

Regiment | Pet |

Dual | Semi-uniform pentagonal prism |

Conjugate | Pentagrammic tegum |

Abstract & topological properties | |

Flag count | 60 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×A_{1}, order 20 |

Flag orbits | 3 |

Convex | Yes |

Nature | Tame |

The **pentagonal tegum** (OBSA: **pet**) also called a **pentagonal bipyramid** or **pentagonal dipyramid**, is one of the 92 Johnson solids (J_{13}). It has 10 equilateral triangles as faces, with 2 order-5 and 5 order-4 vertices. It can be constructed by joining two pentagonal pyramids at their bases

It is one of three regular polygonal tegums to be CRF. The others are the regular octahedron (square tegum) and the triangular tegum.

The pentagonal tegum (with theoretical edge length 1) is the vertex figure of the pentagonal duotegum, which cannot be made uniform, because the Johnson solid variant is not circumscribable.

## Vertex coordinates[edit | edit source]

A pentagonal tegum of edge length 1 has the following vertices:

- ,
- ,
- ,
- .

## Representations[edit | edit source]

A pentagonal tegum has the following Coxeter diagrams:

- oxo5ooo&#xt
- yo ox5oo&#zx (y = )

## Variations[edit | edit source]

The pentagonal tegum can have the height of its pyramids varied while maintaining its full symmetry. These variations generally have 10 isosceles triangles for faces.

One notable variations can be obtained as the dual of the uniform pentagonal prism, which can be represented by m2m5o. In this variant the side edges are times the length of the edges of the base pentagon, and all the dihedral angles are . Each face has apex angle and base angles . If the base pentagon has edge length 1, its height is .

A pentagonal tegum with base edges of length b and side edges of length l has volume given by .

## Related polyhedra[edit | edit source]

A pentagonal prism can be inserted between the halves of the pentagonal bipyramid to produce the elongated pentagonal bipyramid. if a pentagonal antiprism is inserted instead, the result is the gyroelongated pentagonal bipyramid, better known as the regular icosahedron.

## External links[edit | edit source]

- Klitzing, Richard. "pedpy".
- Quickfur. "The Pentagonal Bipyramid".

- Wikipedia contributors. "Pentagonal bipyramid".
- McCooey, David. "Pentagonal Dipyramid J13"