| Triangular bipyramid | |
|---|---|
 | |
| Type | Bipyramid Deltahedra Johnson J11 –J12 –J13 Simplicial |
| Faces | 6triangles |
| Edges | 9 |
| Vertices | 5 |
| Vertex configuration | |
| Symmetry group | |
| Dihedral angle (degrees) | As a Johnson solid:
|
| Dual polyhedron | triangular prism |
| Properties | convex, composite (Johnson solid), face-transitive |
Atriangular bipyramid is ahexahedron, a polyhedron with six triangular faces. It is constructed by attaching twotetrahedra face-to-face. The same shape is also known as atriangular dipyramid[1][2] ortrigonal bipyramid.[3] If these tetrahedra are regular, all faces of a triangular bipyramid areequilateral. It is an example of adeltahedron,composite polyhedron, andJohnson solid.
Many polyhedra are related to the triangular bipyramid, such as similar shapes derived from different approaches and thetriangular prism as itsdual polyhedron. Applications of a triangular bipyramid includetrigonal bipyramidal molecular geometry, which describes itsatom cluster, a solution of theThomson problem, and the representation ofcolor order systems by the eighteenth century.
Like otherbipyramids, a triangular bipyramid can be constructed by attaching two tetrahedra face-to-face.[2] These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, fivevertices, and nine edges.[3] Because of its triangular faces with any type, the triangular bipyramid is asimplicial polyhedron like other infinitely many bipyramids.[4] Aright bipyramid is one in which theapices of both pyramids are on a line passing through the center of the base, such that its faces are isosceles triangles.[5] If two tetrahedra are otherwise, the triangular bipyramid is oblique.[6][7]
According toSteinitz's theorem, agraph can be represented as theskeleton of a polyhedron if it is aplanar (can be drawn without crossing any edges) andthree-connected graph (it remains connected if any two vertices are removed). A triangular bipyramid is represented by a graph with nine edges, constructed by adding one vertex to the vertices of awheel graph representingtetrahedra.[8][9]
Like other right bipyramids, a triangular bipyramid hasthree-dimensional point-group symmetry, thedihedral group of order twelve: the appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle around theaxis of symmetry (a line passing through two vertices and the base's center vertically), and it hasmirror symmetry with any bisector of the base; it is also symmetrical by reflection across a horizontal plane.[10] A triangular bipyramid isface-transitive (or isohedral).[11]
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If the tetrahedra are regular, all edges of a triangular bipyramid are equal in length and formequilateral triangular faces. A polyhedron with only equilateral triangles as faces is called adeltahedron. There are eight convex deltahedra, one of which is a triangular bipyramid withregular polygonal faces.[1] A convex polyhedron in which all of its faces are regular polygons is aJohnson solid. A triangular bipyramid with regular faces is numbered as the twelfth Johnson solid.[12] It is an example of acomposite polyhedron because it is constructed by attaching tworegular tetrahedra.[13][14]
A triangular bipyramid's surface area is six times that of each triangle. Its volume can be calculated by slicing it into two tetrahedra and adding their volume. In the case of edge length, this is:[14]
Thedihedral angle of a triangular bipyramid can be obtained by adding the dihedral angle of two regular tetrahedra. The dihedral angle of a triangular bipyramid between adjacent triangular faces is that of the regular tetrahedron: 70.5 degrees. In an edge where two tetrahedra are attached, the dihedral angle of adjacent triangles is twice that: 141.1 degrees.[15]
Some types of triangular bipyramids may be derived in different ways. TheKleetope of a triangular bipyramid, its Kleetope can be constructed from a triangular bipyramid by attaching tetrahedra to each of its faces, replacing them with three other triangles; the skeleton of the resulting polyhedron represents theGoldner–Harary graph.[16][17] Another type of triangular bipyramid results from cutting off its vertices, a process known astruncation.[18]
Bipyramids are thedual polyhedron ofprisms. This means the bipyramids' vertices correspond to the faces of a prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other; doubling it results in the original polyhedron. A triangular bipyramid is the dual polyhedron of atriangular prism, and vice versa.[19][3] A triangular prism has five faces, nine edges, and six vertices, with the same symmetry as a triangular bipyramid.[3]
TheThomson problem concerns the minimum energy configuration of charged particles on a sphere. A triangular bipyramid is a known solution in the case of five electrons, placing vertices of a triangular bipyramidwithin a sphere.[20] This solution is aided by a mathematically rigorous computer.[21]
Achemical compound'strigonal bipyramidal molecular geometry may be described as theatom cluster of a triangular bipyramid. This molecule has amain-group element without an activelone pair, described by a model which predicts the geometry of molecules known asVSEPR theory.[22] Examples of this structure includephosphorus pentafluoride andphosphorus pentachloride in the gaseousphase.[23]
Incolor theory, the triangular bipyramid was used to represent the three-dimensionalcolor-order system in primary colors. German astronomerTobias Mayer wrote in 1758 that each of its vertices represents a color: white and black are the top and bottom axial vertices, respectively, and the rest of the vertices are red, blue, and yellow.[24][25]