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Concept
Diamant triangulaire
Résumé
In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four.
The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, (J12). As a Johnson solid with all faces equilateral triangles, it is also a deltahedron.
The following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length :
The dual polyhedron of the triangular bipyramid is the triangular prism, with five faces: two parallel equilateral triangles linked by a chain of three rectangles.
Although the triangular prism has a form that is a uniform polyhedron (with square faces), the dual of the Johnson solid form of the bipyramid has rectangular rather than square faces, and is not uniform.
The triangular bipyramid, dt{2,3}, can be in sequence rectified, rdt{2,3}, truncated, {2,3} and alternated (snubbed), {2,3}:
The triangular bipyramid can be constructed by augmentation of smaller ones, specifically two stacked regular octahedra with 3 triangular bipyramids added around the sides, and 1 tetrahedron above and below. This polyhedron has 24 equilateral triangle faces, but it is not a Johnson solid because it has coplanar faces. It is a coplanar 24-triangle deltahedron. This polyhedron exists as the augmentation of cells in a gyrated alternated cubic honeycomb. Larger triangular polyhedra can be generated similarly, like 9, 16 or 25 triangles per larger triangle face, seen as a section of a triangular tiling.
The triangular bipyramid can form a tessellation of space with octahedra or with truncated tetrahedra.
When projected onto a sphere, it resembles a compound of a trigonal hosohedron and trigonal dihedron.
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