Hexagonal prism

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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	D₆, [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.

Since it has eight 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.

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}x{}. The dual of a hexagonal prism is a hexagonal bipyramid.

The symmetry group of a right pentagonal prism is D_6h of order 24. The rotation group is D₆ of order 12.

Volume

As in most prisms, the volume is found by taking the area of the base, with a side length of $a$ , and multiplying it by the height $h$ .

$V={\frac {3{\sqrt {3}}}{2}}a^{2}\times h$

Other images

A translucent model

Use

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

Hexagonal prismatic honeycomb	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 polychora, including:

truncated tetrahedral prism	truncated octahedral prism	Truncated cuboctahedral prism	Truncated icosahedral prism	Truncated icosidodecahedral prism

runcitruncated 5-cell	omnitruncated 5-cell	runcitruncated 16-cell	omnitruncated tesseract

runcitruncated 24-cell	omnitruncated 24-cell	runcitruncated 600-cell	omnitruncated 120-cell

Related polyhedra and tilings

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)							[6,2]⁺, (622)	[1⁺,6,2], (322)	[6,2⁺], (2*3)


{6,2}	t{6,2}	r{6,2}	2t{6,2}=t{2,6}	2r{6,2}={2,6}	rr{6,2}	tr{6,2}	sr{6,2}	h{6,2}	s{2,6}
Uniform duals


V6²	V12²	V6²	V4.4.6	V2⁶	V4.4.6	V4.4.12	V3.3.3.6	V3²	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.

Dimensional family of omnitruncated polyhedra and tilings: *4.6.2n*
Symmetry *n32 [n,3]	Spherical				Euclidean	Hyperbolic
Symmetry *n32 [n,3]	*232 [2,3] D_3h	*332 [3,3] T_d	*432 [4,3] O_h	*532 [5,3] I_h	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]	*∞32 [∞,3]
Coxeter Schläfli	tr{2,3}	tr{3,3}	tr{4,3}	tr{5,3}	tr{6,3}	tr{7,3}	tr{8,3}	tr{∞,3}
Omnitruncated figure
Vertex figure	4.6.4	4.6.6	4.6.8	4.6.10	4.6.12	4.6.14	4.6.16	4.6.∞
Dual figures
Coxeter
Omnitruncated duals
Face configuration	V4.6.4	V4.6.6	V4.6.8	V4.6.10	V4.6.12	V4.6.14	V4.6.16	V4.6.∞

References

Uniform Honeycombs in 3-Space VRML models
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms

External links

Weisstein, Eric W., "Hexagonal prism", MathWorld.
Hexagonal Prism Interactive Model -- works in your web browser

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Symmetry	3	4	5	6	7	8	9	10	11	12
[2n,2] [n,2] [2n,2⁺] [2n⁺,2]
Image
As spherical polyhedra
Image