# Hexagonal prism

Hexagonal prism | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Hip |

Coxeter diagram | x x6o () |

Conway notation | P6 |

Stewart notation | P_{6} |

Elements | |

Faces | 6 squares, 2 hexagons |

Edges | 6+12 |

Vertices | 12 |

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

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 4–4: 120° |

4–6: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 8 |

Level of complexity | 3 |

Related polytopes | |

Army | Hip |

Regiment | Hip |

Dual | Hexagonal tegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 72 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Skeleton | GP(6,1) |

Properties | |

Symmetry | G_{2}×A_{1}, order 24 |

Convex | Yes |

Nature | Tame |

The **hexagonal prism** or **hip** is a prismatic uniform polyhedron. It consists of 2 hexagons and 6 squares. Each vertex joins one hexagon and two squares. As the name suggests, it is a prism based on a hexagon.

It can tile 3D space to form the hexagonal prismatic honeycomb.

## Vertex coordinates[edit | edit source]

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

- ,
- .

## Representations[edit | edit source]

A hexagonal prism has the following Coxeter diagrams:

- x x6o () (full symmetry)
- x x3x () (A2×A1 symmetry, generally a ditrigonal prism)
- s2s6x () (generally a ditrigonal trapezoprism)
- xx6oo&#x (hexagonal frustum)
- xx3xx&#x (ditrigonal frustum)
- xxx xux&#xt (A1×A1 axial, square-first)
- xxxx ohho&#xt (A1×A1, vertex-first)
- xx xu ho&#zx (A1×A1×A1 symmetry)

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

The hexagonal prism has a semi-uniform variant of the form x y6o 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 hexagonal prism with base edges of length a and side edges of length b can be alternated to form a triangular 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 regular octahedron.

## Variations[edit | edit source]

A hexagonal prism has the following variations:

- Ditrigonal prism - prism with ditrigons as bases, and 2 types of rectangles
- Ditrigonal trapezoprism - isogonal with trapezoid sides
- Hexagonal frustum
- Ditrigonal frustum

## Related polyhedra[edit | edit source]

A triangular cupola can be attached to a base of the hexagonal prism to form the elongated triangular cupola. If a second triangular cupola is attached to the other base in the same orientation, the result is the elongated triangular orthobicupola. If the second cupola is rotated 60º from the first cupola the result is the elongated triangular gyrobicupola.

It is also possible to augment square faces of the hexagonal prism with square pyramids. If one square is augmented the result is the augmented hexagonal prism. If a second square, opposite to the first, is augmented the result is the parabiaugmented hexagonal prism. If two non-opposite, non-adjacent squares are augmented the result is the metabiaugmented hexagonal prism. If three mutually non-adjacent squares are augmented the result is the triaugmented hexagonal prism.

Two non-prismatic uniform polyhedron compounds are composed of hexagonal prisms:

There are also an infinite amount of prismatic uniform compounds that are the prisms of compounds of hexagons.

## External links[edit | edit source]

- Klitzing, Richard. "hip".
- Quickfur. "The Hexagonal Prism".

- Wikipedia contributors. "Hexagonal prism".
- McCooey, David. "Hexagonal Prism"

- Hi.gher.Space Wiki Contributors. "Hexagonal prism".