Hexagonal prism

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 $D\_\{6\; \backslash mathrm\{h\}\}$ of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.{{citation

| last1 = Flusser | first1 = J.

| last2 = Suk | first2 = T.

| last3 = Zitofa | first3 = B.

| year = 2017

| title = 2D and 3D Image Analysis by Moments

| publisher = John Wiley & Sons

| isbn = 978-1-119-03935-8

| url = https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126

| page = 126

}}

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:{{citation|title=Geometry|first=Carolyn C.|last=Wheater|publisher=Career Press|year=2007|isbn=9781564149367|pages=236–237}}

$$V\; =\; \backslash frac\{3\; \backslash sqrt\{3\}\}\{2\}a^2h,$$

and its surface area is by summing the area of two regular hexagonal bases and the lateral faces of six squares:

$$S\; =\; 3a(\backslash sqrt\{3\}a+2h).$$

File:Hexagonal prismatic honeycomb.png

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.{{citation|last=Alexandrov|first=A. D.|author-link=Aleksandr Danilovich Aleksandrov|contribution=8.1 Parallelohedra|pages=349–359|publisher=Springer|title=Convex Polyhedra|title-link=Convex Polyhedra (book)|year=2005}} Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.{{citation

| last1 = Delaney | first1 = Gary W.

| last2 = Khoury | first2 = David

| date = February 2013

| doi = 10.1140/epjb/e2012-30445-y

| issue = 2

| journal = The European Physical Journal B

| title = Onset of rigidity in 3D stretched string networks

| volume = 86| page = 44

| bibcode = 2013EPJB...86...44D

}}

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

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

{{reflist}}

• [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space] VRML models
• [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
• [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra [http://www.georgehart.com/virtual-polyhedra/prisms-index.html Prisms and antiprisms]
• {{mathworld | urlname = HexagonalPrism | title = Hexagonal prism}}
• [https://web.archive.org/web/20071008014242/http://polyhedra.org/poly/show/24/hexagonal_prism Hexagonal Prism Interactive Model] -- works in your web browser

Category:Prismatoid polyhedra

Category:Space-filling polyhedra

Category:Zonohedra

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