Editing Triangular bipyramid (section)

=== As a right bipyramid ===
Like other [[bipyramid]]s, a triangular bipyramid can be constructed by attaching two tetrahedra face-to-face.{{r|rajwade}} These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, five [[Vertex (geometry)|vertices]], and nine edges.{{r|king}} A triangular bipyramid is said to be ''right'' if the tetrahedra are symmetrically regular and both of their [[Apex (geometry)|apices]] are on a line passing through the center of the base; otherwise, it is ''oblique''.{{r|niu-xu|alexandrov}}

[[File:Graph of triangular bipyramid.svg|Graph of a triangular bipyramid|alt=A line drawing with multicolored dots|thumb|left|upright]]
According to [[Steinitz's theorem]], a [[Graph (discrete mathematics)|graph]] can be represented as the [[n-skeleton|skeleton]] of a polyhedron if it is a [[Planar graph|planar]] (the edges of the graph do not cross, but intersect at the point) and [[k-vertex-connected graph|three-connected graph]] (one of any two vertices leaves a connected subgraph when removed). A triangular bipyramid is represented by a graph with nine edges, constructed by adding one vertex to the vertices of a [[wheel graph]] representing [[tetrahedra]].{{r|tutte|ssp}}

Like other right bipyramids, a triangular bipyramid has [[Point groups in three dimensions|three-dimensional point-group symmetry]], the [[dihedral group]] <math> D_{3 \mathrm{h}} </math> of order twelve: the appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle around the [[Axial symmetry|axis of symmetry]] (a line passing through two vertices and the base's center vertically), and it has [[mirror symmetry]] with any bisector of the base; it is also symmetrical by reflection across a horizontal plane.{{r|ak}} A triangular bipyramid is [[face-transitive]] (or isohedral).{{r|mclean}}
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