Hexagonal prism

Hexagon prism
Hexagon prism
Type	prism
Symmetry group	prismatic symmetry of order 24
Dual polyhedron	hexagonal bipyramid

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

As a semiregular 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 prismatic symmetry $D_{6\mathrm {h} }$ of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.^[2]

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] $V={\frac {3{\sqrt {3}}}{2}}a^{2}h,$ and its surface area is by summing the area of two regular hexagonal bases and the lateral faces of six squares: $S=3a({\sqrt {3}}a+2h).$

As a parallelohedron

The hexagonal prism is one of the parallelohedron, a polyhedral class that can be translated without rotations in Euclidean space, producing honeycombs; this class was discovered by Evgraf Fedorov in accordance with his studies of crystallography systems. The hexagonal prism is generated from four line segments, three of them parallel to a common plane and the fourth not.^[4] Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.^[5]

The hexagonal prism also exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

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:

References

^ Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN 9780520030565.
^ Flusser, J.; Suk, T.; Zitofa, B. (2017), 2D and 3D Image Analysis by Moments, John Wiley & Sons, p. 126, ISBN 978-1-119-03935-8
^ Wheater, Carolyn C. (2007), Geometry, Career Press, pp. 236–237, ISBN 9781564149367
^ Alexandrov, A. D. (2005), "8.1 Parallelohedra", Convex Polyhedra, Springer, pp. 349–359
^ Delaney, Gary W.; Khoury, David (February 2013), "Onset of rigidity in 3D stretched string networks", The European Physical Journal B, 86 (2): 44, Bibcode:2013EPJB...86...44D, doi:10.1140/epjb/e2012-30445-y

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] Flusser, J.; Suk, T.; Zitofa, B. (2017), 2D and 3D Image Analysis by Moments, John Wiley & Sons, p. 126, ISBN 978-1-119-03935-8

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

[alexandrov-4] Alexandrov, A. D. (2005), "8.1 Parallelohedra", Convex Polyhedra, Springer, pp. 349–359

[onset-5] Delaney, Gary W.; Khoury, David (February 2013), "Onset of rigidity in 3D stretched string networks", The European Physical Journal B, 86 (2): 44, Bibcode:2013EPJB...86...44D, doi:10.1140/epjb/e2012-30445-y

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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