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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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| |}
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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==
| | When you go to shop for your future new apparel, look at how much time, money, and energy you are throwing away. Instead, try online clothes ordering. It's a quick, easy, and fun way to check for your next apparel. There are few reasons should not try it.<br><br>I are unsure if it's very a culture thing but women in the market to lie to themselves a yard. We lie on the size we are, our age, and also forget on what many men we've slept with typically the past.<br><br>Faux leather is completely synthetic. The rii the appearance of real leather, but without the fragility. Since it is synthetic, it is durable and cheap. Is actually important to often for the leather furniture. It can often be found in clothing as well, rendering it a budget-friendly choice for any who want the appearance of leather, but not the rate.<br><br>What could be the best strategy warm up for just about every day of mountain biking? Start hiking more slowly than your normal schedule. Then stop to stretch a person have hiked for fifteen or twenty or so minutes. Stretching cold can injure muscles.<br><br>Learning a brand new skill is for youngsters with PDD-NOS. Their developmental skills challenge them, but right before don't like change. This include more than a few things pertaining to instance potty training, brushing their teeth or combing their head of hair. Because they are so into routine and doing the same thing, in order to learn something new will change their life-style.<br><br>Want to hike more effectively? Avoid a sizable lunch. In fact, skip lunch altogether and just snack when go, keeping your body fed continually with corn chips, trail mix and other convenient food products.<br><br>Sale Queen. Its been a tradition that at the finish and the starting of the ski season, major sales sprouted. Or if you look into making sure that you can have discounted ski clothing, then in order to hit those outlet eating places. Outlet stores offer top-quality, womens ski apparel clothing at discounted estimates. Just a tip: when choosing color, pick easy to and blend hues regarding black and gray. In that way your clothes will match your skis - no matter color often to is.<br><br>If you cherished this short article and you would like to receive more information pertaining to [http://usmerch.co.uk/ gas monkey garage] kindly take a look at our own page. |
| 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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