Triangular bipyramid

Triangular bipyramid
Type	Bipyramid Johnson J11 – J12 – J13
Faces	6 triangles
Edges	9
Vertices	5
Vertex configuration	${\displaystyle 3\times 3^{2}+6\times 3^{2}}$
Symmetry group	${\displaystyle D_{3h}}$
Dual polyhedron	triangular prism
Properties	convex
Net

In geometry, the triangular bipyramid (also called a triangular dipyramid ^{[1] [2]} or trigonal bipyramid ^[3] ) is the hexahedron with six triangular faces, constructed by attaching two tetrahedrons face-to-face. If these tetrahedrons are regular, all faces of triangular bipyramid are equilateral, and it is an example of a deltahedron and of a Johnson solid. The triangular bipyramid has a graph with its construction involving the wheel graph.

Many polyhedrons related to the triangular bipyramid, such as new kinds of similar shapes derived in different approaches, and its dual the triangular prism. The many applications of triangular bipyramid include the trigonal bipyramid molecular geometry that describes its atom cluster, the solution of Thomson problem, and the representation of color order systems by the eighteenth century.

Construction

The triangular bipyramid is constructed like most other bipyramids, by attaching two tetrahedrons face-to-face. ^[2] This construction involves the removal of their base and attaching them, resulting in six triangles, five vertices, and nine edges. ^[3] The triangular bipyramid is said to be right if the tetrahedrons are symmetrically regular and both of their apices are on the line passing through the center of base; otherwise, it is oblique. ^{[4] [5]} If the tetrahedrons are regular, then all edges of the triangular bipyramid are equal in length, making up the faces are equilateral triangles; this is an example of deltahedron. ^[1] More generally, the Johnson solids are the convex polyhedron in which all of the faces are regular, and every convex deltahedra is a Johnson solid. The triangular bipyramid with the regular faces is one of them, enumerated as ${\displaystyle J_{12}}$, the twelfth Johnson solid. ^[6]

Properties

A triangular bipyramid's surface area is six times that of all triangles. In the case of edge length ${\displaystyle a}$, its surface area is: ^[7]

$${\displaystyle {\frac {3{\sqrt {3}}}{2}}a^{2}\approx 2.598a^{2},}$$

Its volume can be calculated by slicing it into two tetrahedras and adding up their volume. In the case of edge length ${\displaystyle a}$, this is: ^[7]

$${\displaystyle {\frac {\sqrt {2}}{6}}a^{3}\approx 0.238a^{3},}$$

3D of a triangular bipyramid

The triangular bipyramid has three-dimensional point group symmetry, the dihedral group ${\displaystyle D_{3h}}$ of order six: the appearance of the 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 base's center vertically), and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane. The dihedral angles of a triangular bipyramid can be calculated by adding the angle of two regular tetrahedra: the angle of tetrahedron between adjacent triangular faces itself is ${\displaystyle \arccos(1/3)\approx 70.5^{\circ }}$, and the dihedral angle of adjacent triangles, on the edge where two tetrahedra attaching is approximately twice of that: ^[8]

$${\displaystyle \arccos \left({\frac {1}{3}}\right)+\arccos \left({\frac {1}{3}}\right)\approx 141.1^{\circ }.}$$

Graph

Graph of triangular bipyramid

According to Steinitz's theorem, a graph can be represented as the skeleton of a polyhedron if it is planar and 3-connected graph. In other words, the edges of that graph do not cross but only intersect at the point, and one of any two vertices leaves a connected subgraph when removed. The triangular bipyramid is represented by a graph with nine edges, constructed by adding one vertex connected other three vertices of wheel graph ${\displaystyle W_{4}}$, where ${\displaystyle W_{n}}$ represents the graph of pyramid with ${\displaystyle n}$-sided polygonal base. ^{[9] [10]}

Sajjad, Sardar & Pan (2024) constructed a chain of triangular bipyramid graphs by arranging them linearly, as in the illustration below. The resistance distance (measurement of two vertices of a graph using the electrical network) of such construction can be computed by applying the series and parallel principles, star-mesh transformation, and Y-Δ transformation. Its structure is an example of the metal-organic frameworks study. ^[11]

Related polyhedra

The Goldner–Harary graph represents the triangular bipyramid attached by tetrahedron.

Some types of triangular bipyramids may be derived in different ways. For example, the Kleetope of polyhedra is a construction involving the attachment of pyramids; in the case of the triangular bipyramid, its Kleetope is constructed by gluing tetrahedrons onto its faces, and its skeleton represents the Goldner–Harary graph by Steinitz's theorem. ^{[12] [13]} Another type of triangular bipyramid is by cutting off all of its vertices; this process is known as truncation. ^[14]

The bipyramids are dual of prisms, for which the bipyramids vertices correspond to the faces of the prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Consequently, the dualization of a dual polyhedron is the original polyhedron itself. Hence, the triangular bipyramid is the dual polyhedron of the triangular prism, and the triangular prism is the dual polyhedron of the triangular bipyramid. ^{[15] [3]} The triangular prism has five faces, nine edges, and six vertices, and its symmetry is shared with its dual. ^[3]

Applications

The known solution of Thomson problem, with one of them is triangular bipyramid.

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. ^[16] This solution is aided by the mathematically rigorous computer. ^[17]

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. ^[18] Some examples of this structure are the phosphorus pentafluoride and phosphorus pentachloride in the gas phase. ^[19]

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, respectively, the top and bottom vertices, whereas the rest of the vertices are red, blue, and yellow. ^{[20] [21]}

References

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