# Pentagonal pyramid

Pentagonal pyramid | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Peppy |

Coxeter diagram | ox5oo&#x |

Conway notation | Y5 |

Stewart notation | Y_{5} |

Elements | |

Faces | 5 triangles, 1 pentagon |

Edges | 5+5 |

Vertices | 1+5 |

Vertex figures | 1 pentagon, edge length 1 |

5 isosceles triangles, edge lengths 1, 1, (1+√5)/2 | |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3-3: |

3-5: | |

Height | |

Central density | 1 |

Number of external pieces | 6 |

Level of complexity | 4 |

Related polytopes | |

Army | Peppy |

Regiment | Peppy |

Dual | Pentagonal pyramid |

Conjugate | Pentagrammic pyramid |

Abstract & topological properties | |

Flag count | 40 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Flag orbits | 4 |

Convex | Yes |

Net count | 15 |

Nature | Tame |

The **pentagonal pyramid** (OBSA: **peppy**) is a pyramid with a pentagonal base and 5 triangles as sides. The version with equilateral triangles as sides is the second of the 92 Johnson solids (J_{2}). In what follows, unless otherwise specified, this what will be meant by a "pentagonal pyramid", even though other variants with isosceles triangles as sides exist.

It is the vertex-first cap of the icosahedron. A regular icosahedron can be constructed by attaching two pentagonal pyramids to the bases of a pentagonal antiprism.

It is one of three regular polygonal pyramids to be CRF. The others are the regular tetrahedron (triangular pyramid) and the square pyramid.

## Vertex coordinates[edit | edit source]

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

- ,
- ,
- .

These coordinates are obtained as a subset of the vertices of the regular icosahedron.

Alternatively, starting from the coordinates of a regular pentagon in the plane, we obtain the pyramid with the following coordinates:

- ,
- ,
- ,
- .

## Related polyhedra[edit | edit source]

Two pentagonal pyramids can be attached at their bases to form a pentagonal tegum.

A pentagonal prism can be attached to the base of a pentagonal pyramid to form the elongated pentagonal pyramid. If a pentagonal antiprism is attached instead, the result is the gyroelongated pentagonal pyramid.

## General variant[edit | edit source]

For the general pentagonal pyramid with base edges of length b and lacing edges of length l, its height is given by , its circumradius by , and its volume is given by .

Pentagonal pyramids occur as vertex figures of 6 uniform polychora, including the convex truncated hexacosichoron, truncated great hecatonicosachoron, truncated great faceted hexacosichoron, quasitruncated great stellated hecatonicosachoron, icosahedral prism, and small stellated dodecahedral prism.

## External links[edit | edit source]

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#4 under ike).

- Klitzing, Richard. "peppy".
- Quickfur. "The Pentagonal Pyramid".

- Weisstein, Eric W. "Pentagonal Pyramid" ("Johnson solid") at MathWorld.

- Wikipedia contributors. "Pentagonal pyramid".
- McCooey, David. "Pentagonal Pyramid"
- Hi.gher.Space Wiki Contributors. "Pentagonal pyramid".