Triangular bipyramid

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	${\displaystyle D_{3h}}$
Dual polyhedron	triangular prism
Properties	convex
Net

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

Many polyhedrons are related to the triangular bipyramid, such as new similar shapes derived in different approaches, and the

, and every convex deltahedron is a Johnson solid. The triangular bipyramid with the regular faces is among numbered the Johnson solids as ${\displaystyle J_{12}}$, the twelfth Johnson solid.

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]

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

${\displaystyle a}$

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

The triangular bipyramid has three-dimensional point group symmetry, the dihedral group ${\displaystyle D_{3h}}$ of order twelve: the appearance of the triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle around the

of a triangular bipyramid with regular faces 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 twice that:

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

Graph

According to

${\displaystyle W_{4}}$, where ${\displaystyle W_{n}}$ represents the graph of

${\displaystyle n}$-

Related polyhedra

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 can be constructed from triangular bipyramid by attaching tetrahedrons onto each of its faces, covering and replacing them with other three triangles; the skeleton of resulting polyhedron represents the Goldner–Harary graph.^[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 the dual polyhedron 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 vice versa.^[15][3] The triangular prism has five faces, nine edges, and six vertices, and it has the same symmetry as the triangular bipyramid.^[3]

Applications

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

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