# Hexagonal prism

In geometry, the **hexagonal prism** is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices.[1]

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

Type | Prismatic uniform polyhedron |

Elements | F = 8, E = 18, V = 12 (χ = 2) |

Faces by sides | 6{4}+2{6} |

Schläfli symbol | t{2,6} or {6}x{} |

Wythoff symbol | 2 6 | 2 2 2 3 | |

Coxeter diagrams | |

Symmetry | D_{6h}, [6,2], (*622), order 24 |

Rotation group | D_{6}, [6,2]^{+}, (622), order 12 |

References | U_{76(d)} |

Dual | Hexagonal dipyramid |

Properties | convex, zonohedron |

Vertex figure 4.4.6 |

Since it has 8 faces, it is an octahedron. However, the term *octahedron* is primarily used to refer to the *regular octahedron*, which has eight triangular faces. Because of the ambiguity of the term *octahedron* and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification.

Before sharpening, many pencils take the shape of a long hexagonal prism.[2]

## As a semiregular (or uniform) polyhedron

If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a **truncated hexagonal hosohedron**, represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}×{}. The dual of a hexagonal prism is a hexagonal bipyramid.

The symmetry group of a right hexagonal prism is *D _{6h}* of order 24. The rotation group is

*D*of order 12.

_{6}## Volume

As in most prisms, the volume is found by taking the area of the base, with a side length of , and multiplying it by the height , giving the formula:[3]

## Symmetry

The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:

Symmetry | D_{6h}, [2,6], (*622) |
C_{6v}, [6], (*66) |
D_{3h}, [2,3], (*322) |
D_{3d}, [2^{+},6], (2*3) | |
---|---|---|---|---|---|

Construction | {6}×{}, |
t{3}×{}, |
s_{2}{2,6}, | ||

Image | |||||

Distortion |

## As part of spatial tesselations

It exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

Hexagonal prismatic honeycomb[1] |
Triangular-hexagonal prismatic honeycomb |
Snub triangular-hexagonal prismatic honeycomb |
Rhombitriangular-hexagonal prismatic honeycomb |

It also exists as cells of a number of four-dimensional uniform 4-polytopes, including:

## Related polyhedra and tilings

Uniform hexagonal dihedral spherical polyhedra | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [6,2], (*622) | [6,2]^{+}, (622) |
[6,2^{+}], (2*3) | ||||||||||||

{6,2} | t{6,2} | r{6,2} | t{2,6} | {2,6} | rr{6,2} | tr{6,2} | sr{6,2} | s{2,6} | ||||||

Duals to uniforms | ||||||||||||||

V6^{2} |
V12^{2} |
V6^{2} |
V4.4.6 | V2^{6} |
V4.4.6 | V4.4.12 | V3.3.3.6 | V3.3.3.3 |

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram *p* < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For *p* > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Sym. * n32[ n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||

*232 [2,3] |
*332 [3,3] |
*432 [4,3] |
*532 [5,3] |
*632 [6,3] |
*732 [7,3] |
*832 [8,3] |
*∞32 [∞,3] |
[12i,3] |
[9i,3] |
[6i,3] |
[3i,3] | |

Figures | ||||||||||||

Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |

Duals | ||||||||||||

Config. | V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |

## See also

Family of uniform prisms | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Polyhedron | |||||||||||

Coxeter | |||||||||||

Tiling | |||||||||||

Config. | 2.4.4 | 3.4.4 | 4.4.4 | 5.4.4 | 6.4.4 | 7.4.4 | 8.4.4 | 9.4.4 | 10.4.4 | 11.4.4 | 12.4.4 |

## References

- Pugh, Anthony (1976),
*Polyheda: A Visual Approach*, University of California Press, pp. 21, 27, 62, ISBN 9780520030565. - Simpson, Audrey (2011),
*Core Mathematics for Cambridge IGCSE*, Cambridge University Press, pp. 266–267, ISBN 9780521727921. - Wheater, Carolyn C. (2007),
*Geometry*, Career Press, pp. 236–237, ISBN 9781564149367.

## External links

- Uniform Honeycombs in 3-Space VRML models
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms
- Weisstein, Eric W. "Hexagonal prism".
*MathWorld*. - Hexagonal Prism Interactive Model -- works in your web browser