Triangular prism

A triangular prism or trigonal prism is a prism with two triangular bases in geometry. If the edges pair with each triangle's vertex and if they are perpendicular to the base, the triangular prism is a right prism. A right triangular prism may be both semiregular and uniform. The triangular prism can be used as the core of constructing other polyhedra, examples are some of the Johnson solids and Schönhardt polyhedron. It has a relationship with the honeycombs and polytopes. It can be found in many real-life applications as in architecture and natural science.

A triangular prism or trigonal prism is a prism with two triangular bases in geometry. If the edges pair with each triangle's vertex and if they are perpendicular to the base, the triangular prism is a right prism. A right triangular prism may be both semiregular and uniform. The triangular prism can be used as the core of constructing other polyhedra, examples are some of the Johnson solids and Schönhardt polyhedron. It has a relationship with the honeycombs and polytopes. It can be found in many real-life applications as in architecture and natural science. V = \frac{1}{2}bh \cdot l. In the case of a right triangular prism, where all its edges are equal in length , its volume can be calculated as the product of the equilateral triangle's area and the distance between bases: V_\text{uniform} = \frac{1}{2} \cdot \frac{\sqrt{3}}{2}l^2 \cdot l = \frac{\sqrt{3}}{4}l^3 \approx 0.433\,l^3 The three-dimensional symmetry group of a triangular prism is prismatic symmetry 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 any prism is a bipyramid, a polyhedron formed by fusing two pyramids base-to-base. In the case of a triangular prism, its dual is a triangular bipyramid, both of which have a common three-dimensional symmetry group. V = \frac{A (h_1 + h_2 + h_3)}{3}. ]] 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. 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. It is named after German mathematician Erich Schönhardt, who described it in 1928, although artist Karlis Johansons exhibited the related structure The triangular prism can be rectified from the convex hull of its midpoints' edges. The resulting construction yields three square faces on the same planes as the square faces of the prism, two equilateral triangle faces on the planes of the triangular ends of the prism, and six more isosceles triangle faces. By giving rise to a graph, its dual forms another convex polyhedron called Herschel enneahedron. The number of triangular prisms can be composed by sharing a common center. Four 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. ==References==

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