Triangular bipyramid

12th Johnson solid; two tetrahedra joined along one face

For the related molecular geometrical structure, see Trigonal bipyramid molecular geometry.

Triangular bipyramid

Type| Bipyramid
Johnson
J11 – J12 – J13
Faces| 6 triangles
Edges| 9
Vertices| 5
Vertex configuration|  3 × 3 2 + 6 × 3 2 {\displaystyle 3\times 3^{2}+6\times 3^{2}} [1]
Symmetry group|  D 3 h {\displaystyle D_{3h}}
Dual polyhedron| triangular prism
Properties| convex
Net

In geometry, the triangular bipyramid[a] is the hexahedron with six triangular faces, constructed by attaching two tetrahedrons face-to-face. All of its faces 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 in various constructions, 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 colors by the eighteenth century.

## Construction[edit]

The triangular bipyramid is constructed like most other bipyramids, by attaching two tetrahedrons face-to-face.[3] This construction involves the removal of their base and attaching them, resulting in six triangles, five vertices, and nine edges.[4] A polyhedra with all faces are equilateral triangles is known as deltahedron. There are eight different convex deltahedra, one of which is the triangular bipyramid.[2] 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 is one of them, enumerated as  J 12 {\displaystyle J_{12}} , the twelfth Johnson solid.[5]

## Properties[edit]

The triangular bipyramid with edge length  a {\displaystyle a} has surface area:

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

six times the area of all equilateral triangles. Its volume:

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

can be calculated by slicing it into two tetrahedra and adding up their volume.[6]

3D of a triangular bipyramid

The triangular bipyramid has three-dimensional point group symmetry, the dihedral group D 3 h {\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; reflecting across a horizontal plane gives the same result. 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  arccos ⁡ ( 1 / 3 ) ≈ 70.5 ∘ {\displaystyle \arccos(1/3)\approx 70.5^{\circ }} , and the dihedral angle of adjacent triangles, on the edge that is attached by two regular tetrahedra is approximately twice of that:[1]

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

## Graph[edit]

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 W 4 {\displaystyle W_{4}} , where  W n {\displaystyle W_{n}} represents the graph of pyramid with  n {\displaystyle n} -sided polygonal base.[7][8]

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.[9]

## Related polyhedra[edit]

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

Some types of triangular bipyramids may be derived based on various constructions. 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.[10][11] Another type is by cutting off all the vertices of a triangular bipyramid; this process is known as truncation.[12]

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.[13][4] The triangular prism has five faces, nine edges, and six vertices, and its symmetry is shared with its dual.[4]

## Applications[edit]

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.[14] This solution is aided by the mathematically rigorous computer.[15]

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.[16] Some examples of this structure are the phosphorus pentafluoride and phosphorus pentachloride in the gas phase.[17]

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.[18][19]

## Notes[edit]

1. ^ It is also referred to as triangular dipyramid,[2][3] or trigonal bipyramid.[4]

## References[edit]

1. ^ a b Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.
2. ^ a b Trigg, Charles W. (1978). "An infinite class of deltahedra". Mathematics Magazine. 51 (1): 55–57. doi:10.1080/0025570X.1978.11976675. JSTOR 2689647. MR 1572246.
3. ^ a b Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84\. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
4. ^ a b c d King, Robert B. (1994). "Polyhedral Dynamics". In Bonchev, Danail D.; Mekenyan, O.G. (eds.). Graph Theoretical Approaches to Chemical Reactivity. Springer. doi:10.1007/978-94-011-1202-4. ISBN 978-94-011-1202-4.
5. ^ Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.
6. ^ Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
7. ^ Tutte, W. T. (2001). Graph Theory. Cambridge University Press. p. 113.
8. ^ Pisanski, Tomaž; Servatius, Brigitte (2013). Configuration from a Graphical Viewpoint. Springer. p. 21\. doi:10.1007/978-0-8176-8364-1. ISBN 978-0-8176-8363-4.
9. ^ Sajjad, Wassid; Sardar, Muhammad S.; Pan, Xiang-Feng (2024). "Computation of resistance distance and Kirchhoff index of chain of triangular bipyramid hexahedron". Applied Mathematics and Computation. 461: 1–12. doi:10.1016/j.amc.2023.128313.
10. ^ Grünbaum, Branko (1967). Convex Polytopes. Wiley Interscience. p. 357.. Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, ISBN 978-0-387-40409-7.
11. ^ Ewald, Günter (1973). "Hamiltonian circuits in simplicial complexes". Geometriae Dedicata. 2 (1): 115–125. doi:10.1007/BF00149287. S2CID 122755203.
12. ^ Haji-Akbari, Amir; Chen, Elizabeth R.; Engel, Michael; Glotzer, Sharon C. (2013). "Packing and self-assembly of truncated triangular bipyramids". Phys. Rev. E. 88 (1): 012127. arXiv:1304.3147. Bibcode:2013PhRvE..88a2127H. doi:10.1103/physreve.88.012127. PMID 23944434. S2CID 8184675..
13. ^ Sibley, Thomas Q. (2015). Thinking Geometrically: A Survey of Geometries. Mathematical Association of American. p. 53.
14. ^ Sloane, N. J. A.; Hardin, R. H.; Duff, T. D. S.; Conway, J. H. (1995), "Minimal-energy clusters of hard spheres", Discrete & Computational Geometry, 14 (3): 237–259, doi:10.1007/BF02570704, MR 1344734, S2CID 26955765
15. ^ Schwartz, Richard Evan (2013). "The Five-Electron Case of Thomson's Problem". Experimental Mathematics. 22 (2): 157–186. doi:10.1080/10586458.2013.766570.
16. ^ Petrucci, R. H.; W. S., Harwood; F. G., Herring (2002). General Chemistry: Principles and Modern Applications (8th ed.). Prentice-Hall. pp. 413–414. ISBN 978-0-13-014329-7. See table 11.1.
17. ^ Housecroft, C. E.; Sharpe, A. G. (2004). Inorganic Chemistry (2nd ed.). Prentice Hall. p. 407\. ISBN 978-0-13-039913-7.
18. ^ Kuehni, Rolf G. (2003). Color Space and Its Divisions: Color Order from Antiquity to the Present. John & Sons Wiley. p. 53.
19. ^ Kuehni, Rolf G. (2013). Color: An Introduction to Practice and Principles. John & Sons Wiley. p. 198\. ISBN 978-1-118-17384-8.

## External links[edit]

* Weisstein, Eric W., "Triangular dipyramid" ("Johnson solid") at MathWorld.
* Conway Notation for Polyhedra Try: dP3

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Johnson solids

Pyramids, cupolae and rotundae|

* square pyramid
* pentagonal pyramid
* triangular cupola
* square cupola
* pentagonal cupola
* pentagonal rotunda

Modified pyramids|

* elongated triangular pyramid
* elongated square pyramid
* elongated pentagonal pyramid
* gyroelongated square pyramid
* gyroelongated pentagonal pyramid
* triangular bipyramid
* pentagonal bipyramid
* elongated triangular bipyramid
* elongated square bipyramid
* elongated pentagonal bipyramid
* gyroelongated square bipyramid

Modified cupolae and rotundae|

* elongated triangular cupola
* elongated square cupola
* elongated pentagonal cupola
* elongated pentagonal rotunda
* gyroelongated triangular cupola
* gyroelongated square cupola
* gyroelongated pentagonal cupola
* gyroelongated pentagonal rotunda
* gyrobifastigium
* triangular orthobicupola
* square orthobicupola
* square gyrobicupola
* pentagonal orthobicupola
* pentagonal gyrobicupola
* pentagonal orthocupolarotunda
* pentagonal gyrocupolarotunda
* pentagonal orthobirotunda
* elongated triangular orthobicupola
* elongated triangular gyrobicupola
* elongated square gyrobicupola
* elongated pentagonal orthobicupola
* elongated pentagonal gyrobicupola
* elongated pentagonal orthocupolarotunda
* elongated pentagonal gyrocupolarotunda
* elongated pentagonal orthobirotunda
* elongated pentagonal gyrobirotunda
* gyroelongated triangular bicupola
* gyroelongated square bicupola
* gyroelongated pentagonal bicupola
* gyroelongated pentagonal cupolarotunda
* gyroelongated pentagonal birotunda

Augmented prisms|

* augmented triangular prism
* biaugmented triangular prism
* triaugmented triangular prism
* augmented pentagonal prism
* biaugmented pentagonal prism
* augmented hexagonal prism
* parabiaugmented hexagonal prism
* metabiaugmented hexagonal prism
* triaugmented hexagonal prism

Modified Platonic solids|

* augmented dodecahedron
* parabiaugmented dodecahedron
* metabiaugmented dodecahedron
* triaugmented dodecahedron
* metabidiminished icosahedron
* tridiminished icosahedron
* augmented tridiminished icosahedron

Modified Archimedean solids|

* augmented truncated tetrahedron
* augmented truncated cube
* biaugmented truncated cube
* augmented truncated dodecahedron
* parabiaugmented truncated dodecahedron
* metabiaugmented truncated dodecahedron
* triaugmented truncated dodecahedron
* gyrate rhombicosidodecahedron
* parabigyrate rhombicosidodecahedron
* metabigyrate rhombicosidodecahedron
* trigyrate rhombicosidodecahedron
* diminished rhombicosidodecahedron
* paragyrate diminished rhombicosidodecahedron
* metagyrate diminished rhombicosidodecahedron
* bigyrate diminished rhombicosidodecahedron
* parabidiminished rhombicosidodecahedron
* metabidiminished rhombicosidodecahedron
* gyrate bidiminished rhombicosidodecahedron
* tridiminished rhombicosidodecahedron

Elementary solids|

* snub disphenoid
* snub square antiprism
* sphenocorona
* augmented sphenocorona
* sphenomegacorona
* hebesphenomegacorona
* disphenocingulum
* bilunabirotunda
* triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)
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