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}×{} |

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 |

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

- As a semiregular (or uniform) polyhedron
- Volume
- Symmetry
- As part of spatial tesselations
- Related polyhedra and tilings
- See also
- References
- External links

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 tilarity 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] }

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

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] }

and it's surface area can be .

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

Name | Regular-hexagonal prism | Hexagonal frustum | Ditrigonal prism | Triambic prism | Ditrigonal trapezoprism |
---|---|---|---|---|---|

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 | |

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:

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 . For *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 |

Family of uniform n-gonal prisms | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Prism name | Digonal prism | (Trigonal) Triangular prism | (Tetragonal) Square prism | Pentagonal prism | Hexagonal prism | Heptagonal prism | Octagonal prism | Enneagonal prism | Decagonal prism | Hendecagonal prism | Dodecagonal prism | ... | Apeirogonal prism |

Polyhedron image | ... | ||||||||||||

Spherical tiling image | Plane tiling image | ||||||||||||

Vertex 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 | ... | ∞.4.4 |

Coxeter diagram | ... |

A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, the **rhombicuboctahedron**, or **small rhombicuboctahedron**, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In geometry, the **truncated octahedron** is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a **6**-zonohedron. It is also the Goldberg polyhedron G_{IV}(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

In geometry, the **truncated cube**, or **truncated hexahedron**, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

In geometry, the **truncated cuboctahedron** is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a **9**-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

In geometry, the **rhombic dodecahedron** is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, the **gyrobifastigium** is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

In Euclidean geometry, **rectification**, also known as **critical truncation** or **complete-truncation** is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a **triangular prism** is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A **right triangular prism** has rectangular sides, otherwise it is *oblique*. A **uniform triangular prism** is a right triangular prism with equilateral bases, and square sides.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

The **bitruncated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, a **quasiregular polyhedron** is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a **snub** is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

In geometry, a **parallelohedron** is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.

In geometry, **chamfering** or **edge-truncation** is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

In geometry, a **plesiohedron** is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy.

- 1 2 Pugh, Anthony (1976),
*Polyhedra: 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 .

- 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

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Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.