Hexagonal prism

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	D₆, [6,2]⁺, (622), order 12
References	U_76(d)
Dual	Hexagonal dipyramid
Properties	convex, zonohedron
Vertex figure 4.4.6

3D model of a uniform hexagonal prism.

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

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]

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 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 $h$ , giving the formula:^[3]

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

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)
Name	Regular-hexagonal prism	Hexagonal frustum	Ditrigonal prism	Triambic prism	Ditrigonal trapezoprism
Construction	{6}×{},		t{3}×{},		s₂{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:

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)	[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

V6²	V12²	V6²	V4.4.6	V2⁶	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* v
Sym. *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Sym. *n32 [n,3]	*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

References[]

^ ^a ^b 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.

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

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[pugh-1] Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN 9780520030565.

[2] Simpson, Audrey (2011), Core Mathematics for Cambridge IGCSE, Cambridge University Press, pp. 266–267, ISBN 9780521727921.

[3] Wheater, Carolyn C. (2007), Geometry, Career Press, pp. 236–237, ISBN 9781564149367.

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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												...