Triangular bipyramid

Short description: Two tetrahedra joined by one face

Triangular bipyramid

Type| Bipyramid
Deltahedra
Johnson
J11 – J12 – J13
Simplicial
Faces| 6 triangles
Edges| 9
Vertices| 5
Vertex configuration| 3×(32)+6×(32)
Symmetry group| D3h
Dihedral angle (degrees)| As a Johnson solid:

* triangle-to-triangle: 70.5°
* triangle-to-triangle if tetrahedra are being attached: 141.1°

Dual polyhedron| triangular prism
Properties| convex,
composite (Johnson solid),
face-transitive

A triangular bipyramid is a hexahedron, a polyhedron with six triangular faces. It is 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.

## Contents

* 1 Special cases
* 1.1 As a right bipyramid
* 1.2 As a Johnson solid
* 2 Related polyhedra
* 3 Applications
* 4 References

## 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] Because of its triangular faces with any type, the triangular bipyramid is a simplicial polyhedron like other infinitely many bipyramids.[4] A right bipyramid is one in which the apices of both pyramids are on a line passing through the center of the base, such that its faces are isosceles triangles.[5] If two tetrahedra are otherwise, the triangular bipyramid is oblique.[6][7]

A line drawing with multicolored dotsGraph 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 (can be drawn without crossing any edges) and three-connected graph (it remains connected if any two vertices are 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.[8][9]

Like other right bipyramids, a triangular bipyramid has three-dimensional point-group symmetry, the dihedral group D3h 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.[10] A triangular bipyramid is face-transitive (or isohedral).[11]

### As a Johnson solid

Triangular bipyramid with regular faces alongside its net

File:J12 triangular bipyramid.stl 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 a Johnson solid. A triangular bipyramid with regular faces is numbered as the twelfth Johnson solid J12.[12] It is an example of a composite polyhedron because it is constructed by attaching two regular tetrahedra.[13][14]

A triangular bipyramid's surface area A is six times that of each triangle. Its volume V can be calculated by slicing it into two tetrahedra and adding their volume. In the case of edge length a, this is:[14] A=332a2≈2.598a2,V=26a3≈0.238a3.

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.[15]

## Related polyhedra

The Kleetope of a triangular bipyramid is obtained by augmenting a triangular bipyramid with tetrahedra on each face.

Some types of triangular bipyramids may be derived in different ways. The Kleetope of a triangular bipyramid, its 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.[16][17] Another type of triangular bipyramid results from cutting off its vertices, a process known as truncation.[18]

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.[19][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.[20] This solution is aided by a mathematically rigorous computer.[21]

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.[22] Examples of this structure include phosphorus pentafluoride and phosphorus pentachloride in the gaseous phase.[23]

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.[24][25]

## References

1. ↑ 1.0 1.1 Trigg, Charles W. (1978). "An infinite class of deltahedra". Mathematics Magazine 51 (1): 55–57. doi:10.1080/0025570X.1978.11976675.
2. ↑ 2.0 2.1 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. https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84.
3. ↑ 3.0 3.1 3.2 3.3 King, Robert B. (1994). "Polyhedral Dynamics". Graph Theoretical Approaches to Chemical Reactivity. Springer. doi:10.1007/978-94-011-1202-4. ISBN 978-94-011-1202-4. https://books.google.com/books?id=c3fsCAAAQBAJ&pg=PA113.
4. ↑ Kumar, C. P. Anil (2020). "On the Coherent Labelling Conjecture of a Polyhedron in Three Dimensions". arXiv:1801.08685 [math.CO].
5. ↑ Montroll, John (2011). Origami Polyhedra Design. CRC Press. p. 6. ISBN 978-1-4398-7106-5. https://books.google.com/books?id=SeTqBgAAQBAJ&pg=PA6. .
6. ↑ Niu, Wenxin; Xu, Guobao (2011). "Crystallographic control of noble metal nanocrystals". Nano Today 6 (3): 265–285. doi:10.1016/j.nantod.2011.04.006.
7. ↑ Alexandrov, Victor (2017). "How many times can the volume of a convex polyhedron be increased by isometric deformations?". Beiträge zur Algebra und Geometrie 58 (3): 549–554. doi:10.1007/s13366-017-0336-8.
8. ↑ Tutte, W. T. (2001). Graph Theory. Cambridge University Press. p. 113. ISBN 978-0-521-79489-3. https://books.google.com/books?id=uTGhooU37h4C&pg=PA113.
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. ↑ Alexander, Daniel C.; Koeberlin, Geralyn M. (2014). Elementary Geometry for College Students (6th ed.). Cengage Learning. p. 403. ISBN 978-1-285-19569-8. https://books.google.com/books?id=EN_KAgAAQBAJ&pg=PA403.
11. ↑ McLean, K. Robin (1990). "Dungeons, dragons, and dice". The Mathematical Gazette 74 (469): 243–256. doi:10.2307/3619822.
12. ↑ 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. https://books.google.com/books?id=51juDwAAQBAJ.
13. ↑ Timofeenko, A. V. (2009). "Convex Polyhedra with Parquet Faces". Doklady Mathematics 80 (2): 720–723. doi:10.1134/S1064562409050238. https://www.interocitors.com/tmp/papers/timo-parquet.pdf.
14. ↑ 14.0 14.1 Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8.
15. ↑ "Convex polyhedra with regular faces". Canadian Journal of Mathematics 18: 169–200. 1966. doi:10.4153/cjm-1966-021-8.
16. ↑ Convex Polytopes. Wiley Interscience. 1967. p. 357. . Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, ISBN 978-0-387-40409-7.
17. ↑ Ewald, Günter (1973). "Hamiltonian circuits in simplicial complexes". Geometriae Dedicata 2 (1): 115–125. doi:10.1007/BF00149287.
18. ↑ 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). doi:10.1103/physreve.88.012127. PMID 23944434. Bibcode: 2013PhRvE..88a2127H. .
19. ↑ Sibley, Thomas Q. (2015). Thinking Geometrically: A Survey of Geometries. Mathematical Association of America. p. 53. ISBN 978-1-939512-08-6. https://books.google.com/books?id=EUh2CgAAQBAJ&pg=PA53.
20. ↑ "Minimal-energy clusters of hard spheres", Discrete & Computational Geometry 14 (3): 237–259, 1995, doi:10.1007/BF02570704
21. ↑ Schwartz, Richard Evan (2013). "The Five-Electron Case of Thomson's Problem". Experimental Mathematics 22 (2): 157–186. doi:10.1080/10586458.2013.766570.
22. ↑ Petrucci, R. H.; Harwood, W. S.; Herring, F. G. (2002). General Chemistry: Principles and Modern Applications (8th ed.). Prentice-Hall. pp. 413–414. ISBN 978-0-13-014329-7. See table 11.1.
23. ↑ Housecroft, C. E.; Sharpe, A. G. (2004). Inorganic Chemistry (2nd ed.). Prentice Hall. p. 407. ISBN 978-0-13-039913-7.
24. ↑ Kuehni, Rolf G. (2003). Color Space and Its Divisions: Color Order from Antiquity to the Present. John & Sons Wiley. p. 53. ISBN 978-0-471-46146-3. https://books.google.com/books?id=2kFVSRGC650C&pg=PA53.
25. ↑ Kuehni, Rolf G. (2013). Color: An Introduction to Practice and Principles. John & Sons Wiley. p. 198. ISBN 978-1-118-17384-8. https://books.google.com/books?id=OD0ogyBwvgwC&pg=PA198.

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* t
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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
* square 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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