Triangular prism

Uniform triangular prism

Type	Prismatic uniform polyhedron
Elements	F = 5, E = 9 V = 6 (χ = 2)
Faces by sides	3{4}+2{3}
Schläfli symbol	t{2,3} or {3}×{}
Wythoff symbol	2 3 \| 2
Coxeter diagram
Symmetry group	D_3h, [3,2], (*322), order 12
Rotation group	D₃, [3,2]⁺, (322), order 6
References	U_76(a)
Dual	Triangular dipyramid
Properties	convex
Vertex figure 4.4.3

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.

3D model of a (uniform) triangular prism

Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.

As a semiregular (or uniform) polyhedronEdit

A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated 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, The dual of a triangular prism is a triangular bipyramid.

The symmetry group of a right 3-sided prism with triangular base is D_3h of order 12. The rotation group is D₃ of order 6. The symmetry group does not contain inversion.

VolumeEdit

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 compute the area of the triangle and multiply this by the length of the prism:

V={\frac {bhl}{2}},

where $b$ is the length of one side of the triangle, $h$ is the length of an altitude drawn to that side, and $l$ is the distance between the triangular faces.

Truncated triangular prismEdit

A truncated right triangular prism has one triangular face truncated (planed) at an oblique angle.^[1]

The volume of a truncated triangular prism with base area A and the three heights h₁, h₂, and h₃ is determined by^[2]

V={\frac {A(h_{1}+h_{2}+h_{3})}{3}}.

FacetingsEdit

There are two full D_3h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C_3v symmetry facetings have one base triangle, 3 lateral crossed square faces, and 3 isosceles triangle lateral faces.

Convex	Facetings
D_3h symmetry			C_3v symmetry

2 {3} 3 {4}	3 {4} 6 ( ) v { }	2 {3} 6 ( ) v { }	1 {3} 3 t'{2} 6 ( ) v { }	1 {3} 3 t'{2} 3 ( ) v { }

Related polyhedra and tilingsEdit

A regular tetrahedron or tetragonal disphenoid can be dissected into two halves with a central square. Each half is a topological triangular prism.

Family of uniform n-gonal prisms v t e
Prism name	Digonal prism	(Trigonal) Triangular prism	(Tetragonal) Square prism	Pentagonal prism	Hexagonal prism	Heptagonal prism	Octagonal prism	Enneagonal prism	Decagonal prism	Hendecagonal prism	Dodecagonal prism	...	Apeirogonal prism
Polyhedron image												...
Spherical tiling image												Plane tiling image
Vertex config.	2.4.4	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4	...	∞.4.4
Coxeter diagram												...

Family of convex cupolae
v
t
e
n	2	3	4	5	6	7	8
Schläfli symbol	${2} \|\| t{2}$	${3} \|\| t{3}$	${4} \|\| t{4}$	${5} \|\| t{5}$	${6} \|\| t{6}$	${7} \|\| t{7}$	${8} \|\| t{8}$
Cupola	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)	Heptagonal cupola (Non-regular face)	Octagonal cupola (Non-regular face)
Related uniform polyhedra	Triangular prism	Cubocta- hedron	Rhombi- cubocta- hedron	Rhomb- icosidodeca- hedron	Rhombi- trihexagonal tiling	Rhombi- triheptagonal tiling	Rhombi- trioctagonal tiling

Symmetry mutationsEdit

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

*n32 symmetry mutation of truncated tilings: t{n,3} v t e
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Symmetry *n32 [n,3]	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Truncated figures
Symbol	t{2,3}	t{3,3}	t{4,3}	t{5,3}	t{6,3}	t{7,3}	t{8,3}	t{∞,3}	t{12i,3}	t{9i,3}	t{6i,3}
Triakis figures
Config.	V3.4.4	V3.6.6	V3.8.8	V3.10.10	V3.12.12	V3.14.14	V3.16.16	V3.∞.∞

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

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paracomp.
Symmetry *n32 [n,3]	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Figure
Config.	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4	3.4.∞.4

CompoundsEdit

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.

HoneycombsEdit

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 polytopesEdit

The triangular prism is first in a dimensional series of semiregular polytopes. 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 simplexes and orthoplexes (equilateral triangles and squares in the case of the triangular prism). In Coxeter's notation the triangular prism is given the symbol −1₂₁.

k₂₁ figures in n dimensional
Space	Finite						Euclidean	Hyperbolic
E_n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
Symmetry	[3^−1,2,1]	[3^0,2,1]	[3^1,2,1]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−1₂₁	0₂₁	1₂₁	2₂₁	3₂₁	4₂₁	5₂₁	6₂₁

Four dimensional spaceEdit

The triangular prism exists as cells of a number of four-dimensional uniform 4-polytopes, including:

Four dimensional polytopes with triangular prisms
Tetrahedral prism	Octahedral prism	Cuboctahedral prism	Icosahedral prism	Icosidodecahedral prism	Truncated dodecahedral prism

Rhomb-icosidodecahedral prism	Rhombi-cuboctahedral prism	Truncated cubic prism	Snub dodecahedral prism	n-gonal antiprismatic prism

Cantellated 5-cell	Cantitruncated 5-cell	Runcinated 5-cell	Runcitruncated 5-cell	Cantellated tesseract	Cantitruncated tesseract	Runcinated tesseract	Runcitruncated tesseract

Cantellated 24-cell	Cantitruncated 24-cell	Runcinated 24-cell	Runcitruncated 24-cell	Cantellated 120-cell	Cantitruncated 120-cell	Runcinated 120-cell	Runcitruncated 120-cell

ReferencesEdit

^ Kern, William F.; Bland, James R. (1938). Solid Mensuration with proofs. p. 81. OCLC 1035479.
^ "Volume of truncated prism". Mathematics Stack Exchange. Retrieved 9 July 2019.

[1] Kern, William F.; Bland, James R. (1938). Solid Mensuration with proofs. p. 81. OCLC 1035479.

[2] "Volume of truncated prism". Mathematics Stack Exchange. Retrieved 9 July 2019.

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