Triangular bipyramid

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.

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

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.

Special cases

As a right bipyramid

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]

Graph of a triangular bipyramid

According to Steinitz's theorem, a graph can be represented as the skeleton of a polyhedron if it is a planar (the edges of the graph do not cross, but intersect at the point) and 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.^[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: the appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle 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]

As a Johnson solid

Triangular bipyramid with regular faces alongside its net

3D model of a triangular bipyramid as a Johnson solid

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]

Related polyhedra

The Goldner–Harary graph represents a triangular bipyramid, augmented by tetrahedra.

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. 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 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]

Applications

The known solution of the Thomson problem, with one a triangular bipyramid

The Thomson 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 bipyramid within a sphere.^[18] This solution is aided by a mathematically rigorous computer.^[19]

A chemical compound's trigonal bipyramidal molecular geometry may be described as the atom cluster of a triangular bipyramid. This molecule has a main-group element without an active lone pair, described by a model which predicts the geometry of molecules known as VSEPR theory.^[20] Examples of this structure include phosphorus pentafluoride and phosphorus pentachloride in the gaseous phase.^[21]

In color theory, the triangular bipyramid was used to represent the three-dimensional color-order system in primary colors. German astronomer Tobias 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.^[22][23]

References

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