View source for Prism (geometry)

{{Short description|Solid with 2 parallel n-gonal bases connected by n parallelograms}}
{{more sources needed|date=March 2025}}
{{Infobox polyhedron
| name          = Set of uniform {{nowrap|{{mvar|n}}-gonal}} prisms
| image         = Hexagonal Prism BC.svg
| caption       = Example: uniform hexagonal prism ({{math|1=''n'' = 6}})
| type          = [[Uniform polyhedron|uniform]] in the sense of [[Semiregular polyhedron|semiregular]] polyhedron
| euler         = 2
| faces         = {{math|2 ''n''}}-sided [[regular polygon]]s<br>
{{mvar|n}} [[square]]s
| edges         = {{math|3''n''}}
| vertices      = {{math|2''n''}}
| vertex_config = {{math|4.4.''n''}}
| schläfli      = {{math|{''n''}×{ } }}<ref>{{cite book
 | last = Johnson | first = N. W | authorlink = Norman Johnson (mathematician)
 | title = Geometries and Transformations
 | year = 2018
 | chapter-url = https://books.google.com/books?id=adBVDwAAQBAJ&pg=PA223
 | chapter = Chapter 11: Finite symmetry groups
 | publisher = Cambridge University Press | isbn = 978-1-107-10340-5}} See 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b.</ref><br />{{math|t{2,''n''} }}
| wythoff       = 
| coxeter       = {{CDD|node_1|2|node_1|n|node}}
| conway        = {{math|P''n''}}
| symmetry      = [[Dihedral symmetry in three dimensions|{{math|D<sub>''n''h</sub>, [''n'',2], (*''n''22),}}]] order {{math|4''n''}}
| rotation_group = {{math|D<sub>''n''</sub>, [''n'',2]<sup>+</sup>, (''n''22),}} order {{math|2''n''}}
| surface_area  = 
| volume        = 
| dual          = [[Convex polytope|convex]] dual-[[Uniform polyhedron|uniform]] {{nowrap|{{mvar|n}}-gonal}} [[bipyramid]]
| properties    = convex, [[regular polygon]] faces, [[Isogonal figure|isogonal]], translated bases, sides ⊥ bases
| vertex_figure = 
| net           = Generalized prisim net.svg
| net_caption   = Example: net of uniform enneagonal prism ({{math|1=''n'' = 9}})
}}

In [[geometry]], a '''prism''' is a [[polyhedron]] comprising an {{nowrap|{{mvar|n}}-sided}} polygon [[Base (geometry)|base]], a second base which is a [[Translation (geometry)|translated]] copy (rigidly moved without rotation) of the first, and {{mvar|n}} other [[Face (geometry)|faces]], necessarily all [[parallelogram]]s, joining [[corresponding sides]] of the two bases. All [[Cross section (geometry)|cross-sections]] parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a [[pentagonal]] base is called a pentagonal prism. Prisms are a subclass of [[prismatoid]]s.<ref name="prismatoid">{{cite journal
 | last = Grünbaum | first = Branko
 | title = Isogonal Prismatoids
 | journal = Discrete & Computational Geometry
 | year = 1997
 | volume = 18
 | pages = 13&ndash;52
 | doi = 10.1007/PL00009307
| doi-access = free
 }}</ref>

Like many basic geometric terms, the word ''prism'' ({{ety|el|''πρίσμα'' (prisma)|something sawed}}) was first used in [[Euclid's Elements|Euclid's ''Elements'']]. Euclid defined the term in Book XI as "a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms". However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers).<ref name="malton-1774">{{cite book
 | last = Malton | first = Thomas
 | title = A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the Mathematics
 | year = 1774
 | url = https://books.google.com/books?id=-3tLfuCB97AC&pg=PA360
 | publisher = author, and sold
 | page = 360}}</ref><ref name="elliot-1845">{{cite book
 | last = Elliot | first = James
 | title = Key to the Complete Treatise on Practical Geometry and Mensuration: Containing Full Demonstrations of the Rules
 | year = 1845
 | url = https://books.google.com/books?id=qilLAAAAYAAJ&pg=PA3
 | publisher = Longman, Brown, Green, and Longmans
 | page = 3}}</ref>

== Oblique vs right ==
An '''oblique prism''' is a prism in which the joining edges and faces are '''''not [[perpendicular]]''''' to the base faces.

Example: a [[parallelepiped]] is an oblique prism whose base is a [[parallelogram]], or equivalently a polyhedron with six parallelogram faces.

[[File:Right Prism.svg|thumb|Right prism]]
A '''''right'' prism''' is a prism in which the joining edges and faces are ''[[perpendicular]]'' to the base faces.<ref name="mensuration">{{cite book
 | last1 = Kern | first1 = William F.
 | last2 = Bland | first2 = James R.
 | title = Solid Mensuration with proofs
 | year = 1938
 | pages = 28}}</ref> This applies [[if and only if]] all the joining faces are ''[[rectangular]]''.

The [[Dual polyhedron|dual]] of a ''right'' {{mvar|n}}-prism is a ''right'' {{mvar|n}}-[[bipyramid]].

A right prism (with rectangular sides) with [[Regular polygon|regular {{mvar|n}}-gon]] bases has Schläfli symbol {{math|{ }×{''n''}.}} It approaches a [[Cylinder (geometry)|cylinder]] as {{mvar|n}} approaches [[infinity]].<ref name="cylinder">{{cite book
 | last = Geretschlager | first = Robert 
 | title = Engaging Young Students In Mathematics Through Competitions: World Perspectives And Practices
 | year = 2020
 | url = https://books.google.com/books?id=2O_CDwAAQBAJ&pg=PA39
 | page = 39
 | publisher = [[World Scientific]]
 | volume = 1
 | isbn = 978-981-120-582-8
}}</ref>

=== Special cases ===
*A right rectangular prism (with a rectangular base) is also called a ''[[cuboid]]'', or informally a ''rectangular box''. A right rectangular prism has [[Schläfli symbol]] {{math|{ }×{ }×{ }.}}
*A right square prism (with a square base) is also called a ''square cuboid'', or informally a ''square box''.

Note: some texts may apply the term ''rectangular prism'' or ''square prism'' to both a right rectangular-based prism and a right square-based prism.

== Types ==

=== Regular prism ===
A '''regular prism''' is a prism with [[Regular polygon|regular]] bases.

=== Uniform prism ===
A '''uniform prism''' or '''semiregular prism''' is a right prism with regular bases and all edges of the same length.

Thus all the side faces of a uniform prism are [[square]]s.

Thus all the faces of a uniform prism are regular polygons. Also, such prisms are [[Isogonal figure|isogonal]]; thus they are [[uniform polyhedra]]. They form one of the two infinite series of [[semiregular polyhedra]], the other series being formed by the [[antiprism]]s.

A uniform {{mvar|n}}-gonal prism has [[Schläfli symbol]] {{math|t{2,''n''}.}}

{{UniformPrisms}}

== Properties ==

=== Volume ===
The [[volume]] of a prism is the product of the [[area]] of the base by the height, i.e. the distance between the two base faces (in the case of a non-right prism, note that this means the perpendicular distance).

The volume is therefore:
:<math>V = Bh,</math>

where {{mvar|B}} is the base area and {{mvar|h}} is the height.

The volume of a prism whose base is an {{mvar|n}}-sided [[regular polygon]] with side length {{mvar|s}} is therefore:
<math display=block>V = \frac{n}{4} h s^2 \cot\frac{\pi}{n}.</math>

=== Surface area ===
The surface [[area]] of a right prism is:
:<math>2B + Ph,</math>
where {{mvar|B}} is the area of the base, {{mvar|h}} the height, and {{mvar|P}} the base [[perimeter]].

The surface area of a right prism whose base is a regular {{mvar|n}}-sided [[polygon]] with side length {{mvar|s}}, and with height {{mvar|h}}, is therefore:

:<math>A = \frac{n}{2} s^2 \cot\frac{\pi}{n} + nsh.</math>

=== Symmetry ===
The [[symmetry group]] of a right {{mvar|n}}-sided prism with regular base is {{math|[[Dihedral group|D<sub>''n''h</sub>]]}} of order {{math|4''n''}}, except in the case of a cube, which has the larger symmetry group {{math|[[Octahedral symmetry|O<sub>h</sub>]]}} of order 48, which has three versions of {{math|D<sub>4h</sub>}} as [[subgroup]]s. The [[Point groups in three dimensions#Rotation groups|rotation group]] is {{math|D<sub>''n''</sub>}} of order {{math|2''n''}}, except in the case of a cube, which has the larger symmetry group {{math|O}} of order 24, which has three versions of {{math|D<sub>4</sub>}} as subgroups.

The symmetry group {{math|D<sub>''n''h</sub>}} contains [[Point reflection|inversion]] [[If and only if|iff]] {{mvar|n}} is even.

The [[Hosohedron|hosohedra]] and [[Dihedron|dihedra]] also possess dihedral symmetry, and an {{mvar|n}}-gonal prism can be constructed via the [[Truncation (geometry)|geometrical truncation]] of an {{mvar|n}}-gonal hosohedron, as well as through the [[cantellation]] or [[Expansion (geometry)|expansion]] of an {{mvar|n}}-gonal dihedron.<!--do hosohedra work with expansion?-->

=== [[Schlegel diagram]]s ===
{| class=wikitable
|- align=center
|[[File:Triangular prismatic graph.svg|100px]]<BR />P3
|[[File:Cubical graph.svg|100px]]<BR />P4
|[[File:Pentagonal prismatic graph.svg|100px]]<BR />P5
|[[File:Hexagonal prismatic graph.svg|100px]]<BR />P6
|[[File:Heptagonal prismatic graph.svg|100px]]<BR />P7
|[[File:Octagonal prismatic graph.svg|100px]]<BR />P8
|}

== Similar polytopes ==

=== Truncated prism ===
[[File:TruncatedTriangularPrism.svg|thumb|left|220px|Example of a truncated triangular prism. Its top face is truncated at an oblique angle, but it is not an [[#Oblique prism|oblique prism]].]]
A '''truncated prism''' is formed when  prism is sliced by a plane that is not [[Parallel (geometry)|parallel]] to its bases. A truncated prism's bases are not [[Congruence (geometry)|congruent]], and its sides are not parallelograms.{{sfnp|Kern|Bland|1938|p=81}}

=== Twisted prism ===
A '''twisted prism''' is a nonconvex polyhedron constructed from a uniform {{mvar|n}}-prism with each side face bisected on the square diagonal, by twisting the top, usually (but not necessarily) by {{sfrac|{{pi}}|''n''}} radians ({{sfrac|180|''n''}} degrees). If the bisectors are slanted to the left, then twisting the top base in the right direction (looking at the top of the prism) by a small angle gives nonconvex polyhedron and twisting it in the left direction, a convex polyhedron (see twisted square prism on the image). If the bisectors are slanted to the right, then twisting the top base in the left direction gives nonconvex polyhedron, in the right direction, convex one (see twisted dodecagonal prism).<ref name="gorini-2003">{{cite book
 | last = Gorini | first = Catherine A.
 | title = The facts on file: Geometry handbook
 | year = 2003
 | url = https://archive.org/details/factsonfilegeome0000gori/page/172/mode/2up
 | isbn = 0-8160-4875-4
 | page = 172| publisher = Facts On File
 }}</ref><ref>{{Cite web|url=http://www.korthalsaltes.com/cuadros.php?type=twisted-prisms|title = Pictures of Twisted Prisms}}</ref>

A twisted prism cannot be [[Dissection (geometry)|dissected]] into tetrahedra without adding new vertices. The simplest twisted prism has triangle bases and is called a [[Schönhardt polyhedron]].

An {{mvar|n}}-gonal ''twisted prism'' is topologically identical to the {{mvar|n}}-gonal uniform [[antiprism]], but has half the [[List of spherical symmetry groups#Dihedral symmetry|symmetry group]]: {{math|D<sub>''n''</sub>, [''n'',2]<sup>+</sup>}}, order {{math|2''n''}}. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles. Any twisted {{mvar|n}}-gonal prism is an antiprism, so the twisted square prism and twisted dodecagonal prism shown on the image are both antiprisms.

{| class=wikitable style="text-align:center; border:none"
!3-gonal
!colspan=2|4-gonal
!12-gonal
|-
|style="border-bottom:none" |[[File:Schönhardt polyhedron.svg|120px]]
|style="border-bottom:none" |[[File:Twisted square antiprism.png|120px]]
|style="border-bottom:none" |[[File:Square antiprism.png|120px]]
|style="border-bottom:none" |[[File:Twisted_dodecagonal_antiprism.png|180px]]
|-
|style="border-top:none" |[[Schönhardt polyhedron]]
|style="border-top:none" |Twisted square prism
|style="border-top:none" |[[Square antiprism]]
|style="border-top:none" |Twisted dodecagonal prism
|}

=== Frustum ===
A [[frustum]] is a similar construction to a prism, with [[trapezoid]] lateral faces and differently sized top and bottom polygons.

[[File:Pentagonal frustum.svg|thumb|220px|Example pentagonal frustum]]

=== Star prism ===
{{Further|Prismatic uniform polyhedron}}
A '''star prism''' is a nonconvex polyhedron constructed by two identical [[star polygon]] faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A ''uniform star prism'' will have [[Schläfli symbol]] {{math|{''p''/''q''} × { },}} with {{mvar|p}} rectangles and 2 {{math|{''p''/''q''} }} faces. It is topologically identical to a {{mvar|p}}-gonal prism.

{| class=wikitable
|+ Examples
!{{math|{ }×{ }<sub>180</sub>×{ } }}
!colspan=2|{{math|[[Truncation (geometry)#Generalized truncations|t<sub>a</sub>]]{3}×{ } }}
!{{math|{5/2}×{ } }}
!{{math|{7/2}×{ } }}
!{{math|{7/3}×{ } }}
!{{math|{8/3}×{ } }}
|- align=center
|{{math|D<sub>2h</sub>}}, order 8
|colspan=2|{{math|D<sub>3h</sub>}}, order 12
|{{math|D<sub>5h</sub>}}, order 20
|colspan=2|{{math|D<sub>7h</sub>}}, order 28
|{{math|D<sub>8h</sub>}}, order 32
|-
|[[File:crossed-square_prism.png|100px]]
|[[File:Crossed hexagonal prism.png|100px]]
|[[File:Crossed-unequal hexagonal prism.png|100px]]
|[[File:Pentagrammic prism.png|100px]]
|[[File:Heptagrammic prism 7-2.png|100px]]
|[[File:Heptagrammic prism 7-3.png|100px]]
|[[File:Prism 8-3.png|100px]]
|}

=== Crossed prism ===
A '''crossed prism''' is a nonconvex polyhedron constructed from a prism, where the vertices of one base are [[Point reflection|inverted around the center]] of this base (or rotated by 180°). This transforms the side rectangular faces into [[crossed rectangle]]s. For a regular polygon base, the appearance is an {{mvar|n}}-gonal [[hour glass]]. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an {{mvar|n}}-gonal prism.

{| class=wikitable
|+ Examples
!{{math|{ }×{ }<sub>180</sub>×{ }<sub>180</sub>}}
!colspan=2|{{math|t<sub>a</sub>{3}×{ }<sub>180</sub>}}
!{{math|{3}×{ }<sub>180</sub>}}
!{{math|{4}×{ }<sub>180</sub>}}
!{{math|{5}×{ }<sub>180</sub>}}
!{{math|{5/2}×{ }<sub>180</sub>}}
!{{math|{6}×{ }<sub>180</sub>}}
|- align=center
|{{math|D<sub>2h</sub>}}, order 8
|colspan=3|{{math|D<sub>3d</sub>}}, order 12
|{{math|D<sub>4h</sub>}}, order 16
|colspan=2|{{math|D<sub>5d</sub>}}, order 20
|{{math|D<sub>6d</sub>}}, order 24
|-
|[[File:crossed_crossed-square_prism.png|100px]]
|[[File:Crossed crossed-hexagonal prism.png|100px]]
|[[File:Crossed crossed-unequal hexagonal prism.png|100px]]
|[[File:Crossed triangular prism.png|100px]]
|[[File:Crossed cube.png|100px]]
|[[File:Crossed pentagonal prism.png|100px]]
|[[File:Crossed pentagrammic prism.png|100px]]
|[[File:Crossed2 hexagonal prism.png|100px]]
|}

=== Toroidal prism ===
A '''toroidal prism''' is a nonconvex polyhedron like a ''crossed prism'', but without bottom and top base faces, and with simple rectangular side faces closing the polyhedron. This can only be done for even-sided base polygons. These are topological tori, with [[Euler characteristic]] of zero. The topological [[polyhedral net]] can be cut from two rows of a [[square tiling]] (with [[vertex configuration]] {{math|4.4.4.4}}): a band of {{mvar|n}} squares, each attached to a [[Rectangle|crossed rectangle]]. An {{mvar|n}}-gonal toroidal prism has {{math|2''n''}} vertices, {{math|2''n''}} faces: {{mvar|n}} squares and {{mvar|n}} crossed rectangles, and {{math|4''n''}} edges. It is topologically [[Self-dual polyhedron|self-dual]].

{| class=wikitable
|+ Examples
|- align=center
|{{math|D<sub>4h</sub>}}, order 16
|{{math|D<sub>6h</sub>}}, order 24
|- align=center
|{{math|1=''V'' = 8}}, {{math|1=''E'' = 16}}, {{math|1=''F'' = 8}}
|{{math|1=''V'' = 12}}, {{math|1=''E'' = 24}}, {{math|1=''F'' = 12}}
|-
|[[File:Toroidal_square_prism.png|100px]]
|[[File:Toroidal hexagonal prism.png|100px]]
|}

=== Prismatic polytope ===
A ''prismatic [[polytope]]'' is a higher-dimensional generalization of a prism. An {{mvar|n}}-dimensional prismatic polytope is constructed from two ({{math|''n'' − 1}})-dimensional polytopes, translated into the next dimension.

The prismatic {{mvar|n}}-polytope elements are doubled from the ({{math|''n'' − 1}})-polytope elements and then creating new elements from the next lower element.

Take an {{mvar|n}}-polytope with {{mvar|F<sub>i</sub>}} [[Face (geometry)|{{mvar|i}}-face]] elements ({{math|''i'' {{=}} 0, ..., ''n''}}). Its ({{math|''n'' + 1}})-polytope prism will have {{math|2''F<sub>i</sub>'' + ''F''<sub>''i''−1</sub>}} {{mvar|i}}-face elements. (With {{math|''F''<sub>−1</sub> {{=}} 0}}, {{math|''F<sub>n</sub>'' {{=}} 1}}.)

By dimension:
*Take a [[polygon]] with {{mvar|n}} vertices, {{mvar|n}} edges. Its prism has {{math|2''n''}} vertices, {{math|3''n''}} edges, and {{math|2 + ''n''}} faces.
*Take a [[polyhedron]] with {{mvar|V}} vertices, {{mvar|E}} edges, and {{mvar|F}} faces. Its prism has {{math|2''V''}} vertices, {{math|2''E'' + ''V''}} edges, {{math|2''F'' + ''E''}} faces, and {{math|2 + ''F''}} cells.
*Take a [[polychoron]] with {{mvar|V}} vertices, {{mvar|E}} edges, {{mvar|F}} faces, and {{mvar|C}} cells. Its prism has {{math|2''V''}} vertices, {{math|2''E'' + ''V''}} edges, {{math|2''F'' + ''E''}} faces,  {{math|2''C'' + ''F''}} cells, and {{math|2 + ''C''}} hypercells.

=== Uniform prismatic polytope ===
{{See also|Uniform 4-polytope#Prismatic_uniform 4-polytopes}}
{{See also|Uniform 5-polytope#Uniform_prismatic forms}}
A regular {{mvar|n}}-polytope represented by [[Schläfli symbol]] {{math|{''p'',''q'',...,''t''} }} can form a uniform prismatic ({{math|''n'' + 1}})-polytope represented by a [[Cartesian product]] of [[Schläfli symbol#Prismatic forms|two Schläfli symbols]]: {{math|{''p'',''q'',...,''t''}×{ }.}}

By dimension:
*A 0-polytopic prism is a [[line segment]], represented by an empty [[Schläfli symbol]] {{math|{ }.}}
*:[[Image:Complete graph K2.svg|60px]]
*A 1-polytopic prism is a [[rectangle]], made from 2 translated line segments. It is represented as the product Schläfli symbol {{math|{ }×{ }.}} If it is [[square]], symmetry can be reduced: {{math|{ }×{ } {{=}} {4}.}}
*:Example: [[Image:Square diagonals.svg|60px]], Square, {{math|{ }×{ },}} two parallel line segments, connected by two line segment ''sides''.
*A [[polygon]]al prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon {{math|{''p''} }} can construct a uniform {{mvar|n}}-gonal prism represented by the product {{math|{''p''}×{ }.}} If {{math|''p'' {{=}} 4}}, with square sides symmetry it becomes a [[cube]]: {{math|{4}×{ } {{=}} {4,3}.}}
*:Example: [[Image:Pentagonal prism.png|60px]], [[Pentagonal prism]], {{math|{5}×{ },}} two parallel [[pentagon]]s connected by 5 rectangular ''sides''.
*A [[Polyhedron|polyhedral]] prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron {{math|{''p'',''q''} }} can construct the uniform polychoric prism, represented by the product {{math|{''p'',''q''}×{ }.}} If the polyhedron and the sides are cubes, it becomes a [[tesseract]]: {{math|1={4,3}×{ } = {4,3,3}.}}
*:Example: [[Image:Dodecahedral prism.png|50px]], [[Dodecahedral prism]], {{math|{5,3}×{ },}} two parallel [[dodecahedra]] connected by 12 pentagonal prism ''sides''.
*...
[[File:23%2C29-duoprism_stereographic_closeup.jpg|thumb|A {{math|{23}×{29} }} duoprism, showing edges in [[stereographic projection]]. The squares make a 23×29 grid [[flat torus]].]]
Higher order prismatic polytopes also exist as [[cartesian product]]s of any two or more polytopes. The dimension of a product polytope is the sum of the dimensions of its elements. The first examples of these exist in 4-dimensional space; they are called [[duoprism]]s as the product of two polygons in 4-dimensions.

Regular duoprisms are represented as {{math|{''p''}×{''q''},}} with {{mvar|pq}} vertices, {{math|2''pq''}} edges, {{mvar|pq}} square faces, {{mvar|p}} {{mvar|q}}-gon faces, {{mvar|q}} {{mvar|p}}-gon faces, and bounded by {{mvar|p}} {{mvar|q}}-gonal prisms and {{mvar|q}} {{mvar|p}}-gonal prisms.

For example, {{math|{4}×{4},}} a ''4-4 duoprism'' is a lower symmetry form of a [[tesseract]], as is {{math|{4,3}×{ },}} a ''cubic prism''. {{math|{4}×{4}×{ } }} (4-4 duoprism prism), {{math|{4,3}×{4} }} (cube-4 duoprism) and {{math|{4,3,3}×{ } }} (tesseractic prism) are lower symmetry forms of a [[5-cube]].

== See also ==
*[[Apeirogonal prism]]
*[[Equiprojective polyhedra]]
*[[Rectified prism]]
*[[Prismanes]]
*[[List of shapes]]

== References ==
{{Reflist}}
* {{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

== External links ==
{{EB1911 poster|Prism<!--|volume=22|page=361-->}}
*{{MathWorld |urlname=Prism |title=Prism}}
*[http://www.korthalsaltes.com/selecion.php?sl=selecion3 Paper models of prisms and antiprisms] Free nets of prisms and antiprisms
*[https://web.archive.org/web/20141019220935/http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms] Using nets generated by ''[[Stella (software)|Stella]]''

{{Polyhedron navigator}}
{{Authority control}}

[[Category:Prismatoid polyhedra]]
[[Category:Uniform polyhedra]]