Triangular bipyramid: Difference between revisions

Triangular bipyramid
Type	Bipyramid Deltahedra Johnson J11 – J12 – J13
Faces	6 triangles
Edges	9
Vertices	5
Vertex configuration	${\displaystyle 3\times (3^{2})+6\times (3^{2})}$
Symmetry group	${\displaystyle D_{3\mathrm {h} }}$
Dihedral angle (degrees)	As a Johnson solid: triangle-to-triangle: 70.5° triangle-to-triangle if tetrahedrons being attached: 141.1°
Dual polyhedron	triangular prism
Properties	convex, composite, face-transitive

In geometry, a triangular bipyramid is a hexahedron with six triangular faces constructed by attaching two tetrahedra face-to-face. The same shape is also known as a triangular dipyramid^[1][2] or trigonal bipyramid.^[3] If these tetrahedra are regular, all faces of a triangular bipyramid are equilateral. It is an example of a deltahedron, composite polyhedron, and Johnson solid.

Many polyhedra are related to the triangular bipyramid, such as similar shapes derived from different approaches and the triangular prism as its dual polyhedron. Applications of a triangular bipyramid include trigonal bipyramidal molecular geometry which describes its atom cluster, a solution of the Thomson problem, and the representation of color order systems by the eighteenth century.

Like other bipyramids, 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, five vertices, and nine edges.^[3] A triangular bipyramid is said to be right if the tetrahedra are symmetrically regular and both of their apices are on a line passing through the center of the base; otherwise, it is oblique.^[4][5]

According to Steinitz's theorem, a graph can be represented as the skeleton of a polyhedron if it is a planar and 3-connected graph. The edges of the graph do not cross, but intersect at the point; 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).^[6][7]

Like other right bipyramids, a triangular bipyramid has three-dimensional point-group symmetry: the dihedral group ${\displaystyle D_{3\mathrm {h} }}$ of order twelve.^{[clarification needed]} The appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle^{[clarification needed]} around the 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.^[8] A triangular bipyramid is face-transitive, or isohedral.^[9]

Triangular bipyramid with regular faces alongside its net

If the tetrahedra are regular, all edges of a triangular bipyramid are equal in length and form equilateral triangular faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are eight convex deltahedra, one of which is a triangular bipyramid with regular polygonal faces.^[1] A convex polyhedron in which all of its faces are regular polygons is the Johnson solid, and every convex deltahedron is a Johnson solid. A triangular bipyramid with regular faces is numbered as the twelfth Johnson solid (${\displaystyle J_{12}}$).^[10] It is an example of a composite polyhedron because it is constructed by attaching two regular tetrahedra.^[11][12]

A triangular bipyramid's surface area (${\displaystyle A}$) is six times that of each triangle. Its volume (${\displaystyle V}$) can be calculated by slicing it into two tetrahedra and adding their volume. In the case of edge length (${\displaystyle a}$), this is:^[12] $${\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}a^{2}&\approx 2.598a^{2},\\V&={\frac {\sqrt {2}}{6}}a^{3}&\approx 0.238a^{3}.\end{aligned}}}$$

The dihedral 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.^[13]

Some types of triangular bipyramids may be derived in different ways. The Kleetope of a polyhedron is a construction involving the attachment of pyramids. A triangular bipyramid's 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 the Goldner–Harary graph.^[14][15] Another type of triangular bipyramid results from cutting off its vertices, a process known as truncation.^[16]

Bipyramids are the dual polyhedron of prisms; 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 a triangular prism, and vice versa.^[17][3] A triangular prism has five faces, nine edges and six vertices, with the same symmetry as a triangular bipyramid.^[3]

The Thomson problem concerns the minimum-energy configuration of charged particles on a sphere. One of them is a triangular bipyramid, which is a known solution for the case of five electrons, by placing vertices of a triangular bipyramid inscribed in a sphere.^[18] This solution is aided by the mathematically rigorous computer.^[19]

In the geometry of chemical compound, the trigonal bipyramidal molecular geometry may be described as the atom cluster of the triangular bipyramid. This molecule has a main-group element without an active lone pair, as described by a model that predicts the geometry of molecules known as VSEPR theory.^[20] Some examples of this structure are the phosphorus pentafluoride and phosphorus pentachloride in the gas phase.^[21]

In the study of color theory, the triangular bipyramid was used to represent the three-dimensional color order system in primary color. The German astronomer Tobias Mayer presented in 1758 that each of its vertices represents the colors: white and black are the top and bottom axial vertices respectively, whereas the rest of the vertices are red, blue, and yellow.^[22][23]

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  Graph of a triangular bipyramid
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  The known solution of the Thomson problem, with one a triangular bipyramid
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  The Goldner–Harary graph represents the triangular bipyramid augmented by tetrahedra.