en>David Eppstein: /* Linear involutions */ involutory matrix

2014-05-07T00:26:19Z

en>Rgdboer: /* Linear involutions */ update link

2012-08-08T00:05:12Z

Linear involutions: update link

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{{For|the optical prism|Triangular prism (optics)}}
{{Unreferenced|date=February 2011}}
{{Prism polyhedra db|Prism polyhedron stat table|P3}}
In [[geometry]], a '''triangular prism''' is a three-sided [[Prism (geometry)|prism]]; it is a [[polyhedron]] made of a [[triangle|triangular]] base, a [[Translation (geometry)|translated]] copy, and 3 faces joining corresponding sides.

Equivalently, it is a [[pentahedron]] of which two faces are parallel, while the [[surface normal]]s of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are [[parallelogram]]s. All cross-sections parallel to the base faces are the same triangle.

== As a semiregular (or uniform) polyhedron ==
A right triangular prism is [[semiregular polyhedron|semiregular]] or, more generally, a [[uniform polyhedron]] if the base faces are equilateral [[triangle]]s, and the other three faces are [[square (geometry)|squares]]. It can be seen as a '''[[truncation (geometry)|truncated]] [[hosohedron|trigonal hosohedron]]''', represented by [[Schläfli symbol]] t{2,3}. Alternately it can be seen as the [[Cartesian product]] of a triangle and a [[line segment]], and represented by the product {3}x{}. The [[dual polyhedron|dual]] of a triangular prism is a [[triangular bipyramid]].

The [[symmetry group]] of a right 3-sided prism with triangular base is [[dihedral group|''D3h'']] of order 12. The [[Point groups in three dimensions#Rotation groups|rotation group]] is ''D3'' of order 6. The symmetry group does not contain [[Inversion in a point|inversion]].

== Volume ==
The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to [[Triangle#Computing the area of a triangle|compute the area of the triangle]] and multiply this by the length of the prism:

<math>V = \frac{1}{2} bhl</math>
where b is the triangle base length, h is the triangle height, and l is the length between the triangles.

== Related polyhedra and tilings ==

{{UniformPrisms}}

{{Cupolae}}

This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2n.2n), and [n,3] [[Coxeter group]] symmetry.

{{Truncated figure1 table}}

This polyhedron is topologically related as a part of sequence of [[Cantellation (geometry)|cantellated]] polyhedra with vertex figure (3.4.n.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[vertex-transitive]] figures have (*n32) reflectional [[Orbifold notation|symmetry]].

This polyhedron is topologically related as a part of sequence of [[Cantellation (geometry)|cantellated]] polyhedra with vertex figure (3.4.n.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[vertex-transitive]] figures have (*n32) reflectional [[Orbifold notation|symmetry]].

{{Expanded table}}

=== Compounds ===

There are 4 uniform compounds of triangular prisms:
:[[Compound of four triangular prisms]], [[compound of eight triangular prisms]], [[compound of ten triangular prisms]], [[compound of twenty triangular prisms]].

=== Honeycombs ===
There are 9 uniform honeycombs that include triangular prism cells:
:[[Gyroelongated alternated cubic honeycomb]], [[elongated alternated cubic honeycomb]], [[gyrated triangular prismatic honeycomb]], [[snub square prismatic honeycomb]], [[triangular prismatic honeycomb]], [[triangular-hexagonal prismatic honeycomb]], [[truncated hexagonal prismatic honeycomb]], [[rhombitriangular-hexagonal prismatic honeycomb]], [[snub triangular-hexagonal prismatic honeycomb]], [[elongated triangular prismatic honeycomb]]

=== Related polytopes ===

The triangular prism is first in a dimensional series of [[Uniform k21 polytope|semiregular polytope]]s. Each progressive [[uniform polytope]] is constructed [[vertex figure]] of the previous polytope. [[Thorold Gosset]] identified this series in 1900 as containing all [[regular polytope]] facets, containing all [[simplex]]es and [[orthoplex]]es ([[equilateral triangle]]s and [[square]]s in the case of the triangular prism). In [[Coxeter]]'s notation the triangular prism is given the symbol −121.
{{Gosset_semiregular_polytopes}}

=== Four dimensional space===
The triangular prism exists as cells of a number of four-dimensional [[uniform polychoron|uniform polychora]], including:
{| class=wikitable width=640
|- align=center
|[[tetrahedral prism]] {{CDD|node_1|3|node|3|node|2|node_1}}
|[[octahedral prism]] {{CDD|node_1|3|node|4|node|2|node_1}}
|[[cuboctahedral prism]] {{CDD|node|3|node_1|4|node|2|node_1}}
|[[icosahedral prism]] {{CDD|node_1|3|node|5|node|2|node_1}}
|[[icosidodecahedral prism]] {{CDD|node|3|node_1|5|node|2|node_1}}
|[[Truncated dodecahedral prism]] {{CDD|node|3|node_1|5|node_1|2|node_1}}
|- align=center
|[[File:tetrahedral prism.png|80px]]
|[[File:octahedral prism.png|80px]]
|[[File:cuboctahedral prism.png|80px]]
|[[File:icosahedral prism.png|80px]]
|[[File:icosidodecahedral prism.png|80px]]
|[[File:Truncated dodecahedral prism.png|80px]]
|- align=center
|[[Rhombicosidodecahedral prism|Rhombi-cosidodecahedral prism]] {{CDD|node_1|3|node|5|node_1|2|node_1}}
|[[Rhombicuboctahedral prism|Rhombi-cuboctahedral prism]] {{CDD|node_1|3|node|4|node_1|2|node_1}}
|[[Truncated cubic prism]] {{CDD|node|3|node_1|4|node_1|2|node_1}}
|[[Snub dodecahedral prism]] {{CDD|node_h|5|node_h|3|node_h|2|node_1}}
|[[Uniform antiprismatic prism|n-gonal antiprismatic prism]] {{CDD|node_h|n|node_h|2x|node_h|2|node_1}}
|- align=center
|[[File:Rhombicosidodecahedral prism.png|80px]]
|[[File:Rhombicuboctahedral prism.png|80px]]
|[[File:Truncated cubic prism.png|80px]]
|[[File:Snub dodecahedral prism.png|80px]]
|[[File:Square antiprismatic prism.png|80px]]
|- align=center
|[[Cantellated 5-cell]] {{CDD|node_1|3|node|3|node_1|3|node}}
|[[Cantitruncated 5-cell]] {{CDD|node_1|3|node_1|3|node_1|3|node}}
|[[Runcinated 5-cell]] {{CDD|node_1|3|node|3|node|3|node_1}}
|[[Runcitruncated 5-cell]] {{CDD|node_1|3|node_1|3|node|3|node_1}}
|[[Cantellated tesseract]] {{CDD|node_1|4|node|3|node_1|3|node}}
|[[Cantitruncated tesseract]] {{CDD|node_1|4|node_1|3|node_1|3|node}}
|[[Runcinated tesseract]] {{CDD|node_1|4|node|3|node|3|node_1}}
|[[Runcitruncated tesseract]] {{CDD|node_1|4|node_1|3|node|3|node_1}}
|- align=center
|[[File:4-simplex t02.svg|80px]]
|[[File:4-simplex t012.svg|80px]]
|[[File:4-simplex t03.svg|80px]]
|[[File:4-simplex t013.svg|80px]]
|[[File:4-cube t02.svg|80px]]
|[[File:4-cube t012.svg|80px]]
|[[File:4-cube t03.svg|80px]]
|[[File:4-cube t013.svg|80px]]
|- align=center
|[[Cantellated 24-cell]] {{CDD|node_1|3|node|4|node_1|3|node}}
|[[Cantitruncated 24-cell]] {{CDD|node_1|3|node_1|4|node_1|3|node}}
|[[Runcinated 24-cell]] {{CDD|node_1|3|node|4|node|3|node_1}}
|[[Runcitruncated 24-cell]] {{CDD|node_1|3|node_1|4|node|3|node_1}}
|[[Cantellated 120-cell]] {{CDD|node_1|5|node|3|node_1|3|node}}
|[[Cantitruncated 120-cell]] {{CDD|node_1|5|node_1|3|node_1|3|node}}
|[[Runcinated 120-cell]] {{CDD|node_1|5|node|3|node|3|node_1}}
|[[Runcitruncated 120-cell]] {{CDD|node_1|5|node_1|3|node|3|node_1}}
|- align=center
|[[File:24-cell t02 F4.svg|80px]]
|[[File:24-cell t012 F4.svg|80px]]
|[[File:24-cell t03 F4.svg|80px]]
|[[File:24-cell t013 F4.svg|80px]]
|[[File:120-cell t02 H3.png|80px]]
|[[File:120-cell t012 H3.png|80px]]
|[[File:120-cell t03 H3.png|80px]]
|[[File:120-cell t013 H3.png|80px]]
|}

== See also==
* [[Wedge (geometry)]]

==External links==
* {{mathworld | urlname = TriangularPrism | title = Triangular prism}}
*[http://polyhedra.org/poly/show/22/triangular_prism Interactive Polyhedron: Triangular Prism]
* Whole site dedicated to triangular prisms. Good resource for high school students to learn about how to find the [http://www.triangular-prism.com surface area and volume of a triangular prism] and how to draw its net.

[[Category:Prismatoid polyhedra]]
[[Category:Space-filling polyhedra]]

Affine involution - Revision history

en>David Eppstein: /* Linear involutions */ involutory matrix

en>Rgdboer: /* Linear involutions */ update link