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 | D_{3}, [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.
Equivalently, it is a pentahedron 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.
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 {3}x{}. 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 D_{3} of order 6. The symmetry group does not contain inversion.
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:
where b is the triangle base length, h is the triangle height, and l is the length between the triangles.
It is related to the following sequence of uniform truncated polyhedra and tilings with 3.2n.2n a vertex configuration and [n,3] Coxeter group symmetry.
3.4.4 |
3.6.6 |
3.8.8 |
3.10.10 |
3.12.12 |
3.14.14 |
3.16.16 |
File:Hyperbolic tiling o3x∞x.png 3.∞.∞ |
There are 4 uniform compounds of triangular prisms:
There are 9 uniform honeycombs that include triangular prism cells:
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
The triangular prism exists as cells of a number of four-dimensional uniform polychora, including: