Triangular prism

For the optical prism, see Triangular prism (optics).

Triangular prism
Type	Prism Semiregular polyhedron Uniform polyhedron
Faces	2 triangles 3 squares
Edges	9
Vertices	6
Symmetry group	D3h
Dual polyhedron	Triangular bipyramid

In geometry, a triangular prism or trigonal prism ^[1] is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

The triangular prism can be used in constructing another polyhedron. Examples are some of the Johnson solids, the truncated right triangular prism, and Schönhardt polyhedron.

Properties

A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its bases, and the bases of a triangular prism are triangles. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3 parallelograms as other faces. ^[2] If the prism's edges are perpendicular to the base, the lateral faces are rectangles, and the prism is called a right triangular prism. ^[3] This prism may also be considered a special case of a wedge. ^[4]

3D model of a (uniform) triangular prism

If the base is equilateral and the lateral faces are square, then the right triangular prism is semiregular. A semiregular prism means that the number of its polygonal base's edges equals the number of its square faces. ^[5] More generally, the triangular prism is uniform. This means that a triangular prism has regular faces and has an isogonal symmetry on vertices. ^[6] The three-dimensional symmetry group of a right triangular prism is dihedral group D_3h of order 12: the appearance is unchanged if the triangular prism is rotated one- and two- thirds of a full angle around its axis of symmetry passing through the center's base, and reflecting across a horizontal plane. The dual polyhedron of a triangular prism is a triangular bipyramid. The triangular bipyramid has the same symmetry as the triangular prism. ^[1] The dihedral angle between two adjacent square faces is the internal angle of an equilateral triangle π/3 = 60°, and that between a square and a triangle is π/2 = 90°. ^[7]

The volume of any prism is the product of the area of the base and the distance between the two bases. ^[8] In the case of a triangular prism, its base is a triangle, so its volume can be calculated by multiplying the area of a triangle and the length of the prism:

$${\displaystyle {\frac {bhl}{2}},}$$

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. ^[9] In the case of a right triangular prism, where all its edges are equal in length l, its volume can be calculated as the product of the equilateral triangle's area and length l: ^[10]

$${\displaystyle {\frac {\sqrt {3}}{2}}l^{2}\cdot l\approx 0.433l^{3}}$$

The triangular prism can be represented as the prism graph Π₃. More generally, the prism graph Π_n represents the n-sided prism. ^[11]

Related polyhedron

In construction of polyhedron

Beyond the triangular bipyramid as its dual polyhedron, many other polyhedrons are related to the triangular prism. A Johnson solid is a convex polyhedron with regular faces, and this definition is sometimes omitted uniform polyhedrons such as Archimedean solids, Catalan solids, prisms and antiprisms. ^[12] There are 6 Johnson solids with their construction involving the triangular prism: elongated triangular pyramid, elongated triangular bipyramid, gyrobifastigium, augmented triangular prism, biaugmented triangular prism, and triaugmented triangular prism. The elongated triangular pyramid and the gyroelongated triangular pyramid are constructed by attaching tetrahedron onto the base of a triangular prism. The augmented triangular prism, biaugmented triangular prism, and triaugmented triangular prism are constructed by attaching equilateral square pyramids onto the square face of the prism. The gyrobifastigium is constructed by attaching two triangular prisms along one of its square faces. ^[13]

Truncted right triangular prism

A truncated triangular prism is a triangular prism constructed by truncating its part at an oblique angle. As a result, the two bases are not parallel and every height has a different edge length. If the edges connecting bases are perpendicular to one of its bases, the prism is called a truncated right triangular prism. Given that A is the area of the triangular prism's base, and the three heights h₁, h₂, and h₃, its volume can be determined in the following formula: ^[14]

$${\displaystyle {\frac {A(h_{1}+h_{2}+h_{3})}{3}}.}$$

Schönhardt polyhedron

Schönhardt polyhedron is another polyhedron constructed from a triangular prism with equilateral triangle bases. This way, one of its bases rotates around the prism's centerline and breaks the square faces into skew polygons. Each square face can be re-triangulated with two triangles to form a non-convex dihedral angle. ^[15] As a result, the Schönhardt polyhedron cannot be triangulated by a partition into tetrahedra. It is also that the Schönhardt polyhedron has no internal diagonals. ^[16] It is named after German mathematician Erich Schönhardt, who described it in 1928, although the related structure was exhibited by artist Karlis Johansons in 1921. ^[17]

There are 4 uniform compounds of triangular prisms. They are compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms. ^[18]

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 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 dimensions
Space	Finite						Euclidean	Hyperbolic
En	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter diagram
Symmetry	[3−1,2,1]	[30,2,1]	[31,2,1]	[32,2,1]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
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 space

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

References

Citations

1. King (1994), p.  113.
2. • King (1994), p.  113
   • Berman (1971)
3. Kern & Bland (1938), p. 25.
4. Haul (1893), p.  45.
5. O'Keeffe & Hyde (2020), p.  139.
6. • Berman & Williams (2009), p.  100
   • Messer (2002)
7. Johnson (1966).
8. Kern & Bland (1938), p. 26.
9. • Kinsey, Moore & Prassidis (2011), p.  389
   • Haul (1893), p.  45
10. Berman (1971).
11. Pisanski & Servatius (2013), p.  21.
12. • Todesco (2020), p.  282
    • Williams & Monteleone (2021), p.  23
13. • Rajwade (2001)
    • Berman (1971)
14. Kern & Bland (1938), p. 81.
15. • Schönhardt (1928)
    • Bezdek & Carrigan (2016)
16. Bagemihl (1948).
17. • Schönhardt (1928)
    • Bansod, Nandanwar & Burša (2014)
18. Skilling (1976).

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