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| {{Other uses2|Prism}}
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| {|class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="280"
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| !bgcolor=#e7dcc3 colspan=2|Set of uniform prisms
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| |align=center colspan=2|[[image:Hexagonal Prism BC.svg|220px|Uniform prisms]]<br>(A hexagonal prism is shown)
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| |bgcolor=#e7dcc3|Type||[[uniform polyhedron]]
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| |bgcolor=#e7dcc3|Faces||2+''n'' total:<br>2 [[Regular polygon|{n}]]<br>''n'' [[Square (geometry)|{4}]]
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| |bgcolor=#e7dcc3|Edges||3''n''
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| |bgcolor=#e7dcc3|Vertices||2''n''
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||{n}×{} or ''t''{2, ''n''}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|n|node|2|node_1}}
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| |bgcolor=#e7dcc3|[[Vertex configuration]]||4.4.''n''
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| |bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||[[Dihedral symmetry in three dimensions|D<sub>''n''h</sub>]], [''n'',2], (*''n''22), order 4''n''
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| |bgcolor=#e7dcc3|[[Point groups in three dimensions#Rotation groups|Rotation group]]||D<sub>''n''</sub>, [''n'',2]<sup>+</sup>, (''n''22), order 2''n''
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| |bgcolor=#e7dcc3|[[Dual polyhedron]]||[[bipyramid]]s
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| |bgcolor=#e7dcc3|Properties||convex, semi-regular [[vertex-transitive]]
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| |colspan=2 align=center|[[Image:Generalized prisim net.svg|150px]]<br>[[Net (polyhedron)|''n''-gonal prism net {{nowrap|(''n'' {{=}} 9 here)}}]]
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| [[Image:Azrieli Towers Sept.2007.JPG|thumb|[[Azrieli Towers]] are prism.]]
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| In [[geometry]], a '''prism''' is a [[polyhedron]] with an ''n''-sided [[polygon]]al base, a [[Translation (geometry)|translated]] copy (not in the same plane as the first), and ''n'' other faces (necessarily all [[parallelogram]]s) joining corresponding sides of the two bases. All [[Cross section (geometry)|cross-sections]] parallel to the base faces are the same. Prisms are named for their base, so a prism with a [[pentagonal]] base is called a pentagonal prism. The prisms are a subclass of the [[prismatoid]]s.
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| ==General, right and uniform prisms==
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| A ''right prism'' is a prism in which the joining edges and faces are [[perpendicular]] to the base faces. This applies if the joining faces are [[rectangular]]. If the joining edges and faces are not perpendicular to the base faces, it is called an ''oblique prism''.
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| Some texts may apply the term ''rectangular prism'' or ''square prism'' to both a right rectangular-sided prism and a right square-sided prism.
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| The term ''uniform prism'' can be used for a right prism with square sides, since such prisms are in the set of [[uniform polyhedra]].
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| An ''n''-prism, having [[regular polygon]] ends and [[rectangular]] sides, approaches a [[Cylinder (geometry)|cylindrical]] solid as ''n'' approaches [[infinity]].
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| Right prisms with regular bases and equal edge lengths form one of the two infinite series of [[semiregular polyhedra]], the other series being the [[antiprism]]s.
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| The [[dual polyhedron|dual]] of a right prism is a [[bipyramid]].
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| A [[parallelepiped]] is a prism of which the base is a [[parallelogram]], or equivalently a polyhedron with six faces which are all parallelograms.
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| A right rectangular prism is also called a ''[[cuboid]]'', or informally a ''rectangular box''. A right square prism is simply a ''square box'', and may also be called a ''square cuboid''.
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| ==Volume==
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| The [[volume]] of a prism is the product of the [[area]] of the base and the distance between the two base faces, or the height (in the case of a non-right prism, note that this means the perpendicular distance). | |
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| The volume is therefore:
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| :<math>V = B \cdot h</math>
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| where ''B'' is the base area and ''h'' is the height. The volume of a prism whose base is a regular ''n''-sided [[polygon]] with side length ''s'' is therefore:
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| :<math>V = \frac{n}{4}hs^2 \cot\frac{\pi}{n}.</math>
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| ==Surface area==
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| The surface [[area]] of a right prism is {{nowrap|2 · ''B'' + ''P'' · ''h''}}, where ''B'' is the area of the base, ''h'' the height, and ''P'' the base [[perimeter]].
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| The surface area of a right prism whose base is a regular ''n''-sided [[polygon]] with side length ''s'' and height ''h'' is therefore:
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| :<math>A = \frac{n}{2} s^2 \cot{\frac{\pi}{n}} + n s h.</math> | |
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| ==Symmetry==
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| The [[symmetry group]] of a right ''n''-sided prism with regular base is [[dihedral group|D<sub>''n''h</sub>]] of order 4''n'', except in the case of a cube, which has the larger symmetry group [[octahedral symmetry|O<sub>h</sub>]] of order 48, which has three versions of D<sub>4h</sub> as [[subgroup]]s. The [[Point groups in three dimensions#Rotation groups|rotation group]] is D<sub>''n''</sub> of order 2''n'', except in the case of a cube, which has the larger symmetry group O of order 24, which has three versions of D<sub>4</sub> as subgroups.
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| The symmetry group D<sub>''n''h</sub> contains [[Inversion in a point|inversion]] [[If and only if|iff]] ''n'' is even.
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| ==Prismatic polytope==
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| A ''prismatic [[polytope]]'' is a higher dimensional generalization of a prism. An ''n''-dimensional prismatic polytope is constructed from two ({{nowrap|''n'' − 1}})-dimensional polytopes, translated into the next dimension.
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| The prismatic ''n''-polytope elements are doubled from the ({{nowrap|''n'' − 1}})-polytope elements and then creating new elements from the next lower element.
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| Take an ''n''-polytope with ''f<sub>i</sub>'' [[Face|''i''-face]] elements ({{nowrap|''i'' {{=}} 0, ..., ''n''}}). Its ({{nowrap|''n'' + 1}})-polytope prism will have {{nowrap|2''f''<sub>''i''</sub> + ''f''<sub>''i''−1</sub>}} ''i''-face elements. (With {{nowrap|''f''<sub>−1</sub> {{=}} 0}}, {{nowrap|''f''<sub>''n''</sub> {{=}} 1}}.)
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| By dimension:
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| *Take a [[polygon]] with ''n'' vertices, ''n'' edges. Its prism has 2''n'' vertices, 3''n'' edges, and {{nowrap|2 + ''n''}} faces.
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| *Take a [[polyhedron]] with ''v'' vertices, ''e'' edges, and ''f'' faces. Its prism has 2''v'' vertices, {{nowrap|2''e'' + ''v''}} edges, {{nowrap|2''f'' + ''e''}} faces, and {{nowrap|2 + ''f''}} cells.
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| *Take a [[polychoron]] with ''v'' vertices, ''e'' edges, ''f'' faces and ''c'' cells. Its prism has 2''v'' vertices, {{nowrap|2''e'' + ''v''}} edges, {{nowrap|2''f'' + ''e''}} faces, and {{nowrap|2''c'' + ''f''}} cells, and {{nowrap|2 + ''c''}} hypercells.
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| ===Uniform prismatic polytope===
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| A regular ''n''-polytope represented by [[Schläfli symbol]] {{nowrap|{''p'', ''q'', ...,}} ''t''} can form a uniform prismatic ({{nowrap|''n'' + 1}})-polytope represented by a [[Cartesian product]] of [[Schläfli symbol#Prismatic_forms|two Schläfli symbols]]: {{nowrap|{''p'', ''q'', ...,}} ''t''}×{}.
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| By dimension:
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| *A 0-polytopic prism is a [[line segment]], represented by an empty [[Schläfli symbol]] {}.
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| **[[Image:Complete graph K2.svg|60px]]
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| *A 1-polytopic prism is a [[rectangle]], made from 2 translated line segments. It is represented as the product Schläfli symbol {}×{}. If it is [[Square (geometry)|square]], symmetry can be reduced it: {{nowrap|{}×{} {{=}} {4}.}}
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| **[[Image:Square diagonals.svg|60px]]Example: Square, {}×{}, two parallel line segments, connected by two line segment ''sides''.
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| *A [[polygon]]al prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon {''p''} can construct a uniform ''n''-gonal prism represented by the product {''p''}×{}. If {{nowrap|''p'' {{=}} 4}}, with square sides symmetry it becomes a [[cube]]: {{nowrap|{4}×{} {{=}} {4, 3}.}}
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| **[[Image:Pentagonal prism.png|60px]]Example: [[Pentagonal prism]], {5}×{}, two parallel [[pentagon]]s connected by 5 rectangular ''sides''.
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| *A [[polyhedron|polyhedral]] prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron {''p'', ''q''} can construct the uniform polychoric prism, represented by the product {''p'', ''q''}×{}. If the polyhedron is a cube, and the sides are cubes, it becomes a [[tesseract]]: {4, 3}×{} = {{nowrap|{4, 3, 3}.}}
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| **[[Image:Dodecahedral prism.png|50px]]Example: [[Dodecahedral prism]], {5, 3}×{}, two parallel [[dodecahedron|dodecahedra]] connected by 12 pentagonal prism ''sides''.
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| *...
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| Higher order prismatic polytopes also exist as [[Cartesian product]]s of any two polytopes. The dimension of a polytope is the product of the dimensions of the elements. The first example of these exist in 4-dimensional space are called [[duoprism]]s as the product of two polygons. Regular duoprisms are represented as {''p''}×{''q''}.
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| ==See also==
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| {{UniformPrisms}}
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| *[[Antiprism]]
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| *[[Cylinder (geometry)]]
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| *[[Apeirogonal prism]]
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| == References==
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| * {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }} Chapter 2: Archimedean polyhedra, prisma and antiprisms
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| ==External links==
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| *{{MathWorld |urlname=Prism |title=Prism}}
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| *{{GlossaryForHyperspace |anchor=Prismatic |title=Prismatic polytope}}
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| *[http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms]
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| *[http://www.mathguide.com/lessons/SurfaceArea.html#prisms Surface Area] MATHguide
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| *[http://www.mathguide.com/lessons/Volume.html#prisms Volume] MATHguide
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| *[http://www.korthalsaltes.com/selecion.php?sl=selecion3 Paper models of prisms and antiprisms] Free nets of prisms and antiprisms
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| *[http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms] Using nets generated by ''[[Stella (software)|Stella]]''.
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| *[http://www.software3d.com/Stella.php Stella: Polyhedron Navigator]: Software used to create the 3D and 4D images on this page.
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| {{Polyhedron navigator}}
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| [[Category:Prismatoid polyhedra]]
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| [[Category:Uniform polyhedra]]
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Bryan is usually a celebrity from the building plus the vocation growth last minute tickets very first second to his third hotel record, & , is the resistant. He broken to the picture in 2014 along with his funny blend of downward-house convenience, film legend wonderful appearance and lyrics, is set t inside a major way. The new record Top about the region graph and #2 about the pop maps, generating it the next luke bryan concert tour dates greatest first appearance during luke bryan ticket sale dates (http://lukebryantickets.lazintechnologies.com) those times of 2010 to get a region musician.
The kid of your , is aware of persistence and perseverance are key elements when it comes to a prosperous profession- . His 1st album, Keep Me, generated the Top reaches “All My Pals Say” and “Country Gentleman,” although his energy, Doin’ Issue, identified the singer-three straight No. 8 singles: Different Contacting Is really eagles concert tickets (http://lukebryantickets.pyhgy.com/) a Excellent Point.”
In the slip of 2007, Tour: Bryan & that had a remarkable listing of , which includes Downtown. “It’s much like you’re acquiring a authorization to travel to a higher level, says individuals musicians that have been a part of the Tourover right into a greater level of designers.” It twisted as the best trips in its 10-12 months record.
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