This is a good article. Click here for more information.

Triangular bipyramid

From Wikipedia, the free encyclopedia
(Redirected from Triangular dipyramid)

Template:Short description

Page Module:Infobox/styles.css has no content.

Triangular bipyramid
Lua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
TypeBipyramid
Deltahedra
Johnson
J11 – J12 – J13
Simplicial
Faces6 triangles
Edges9
Vertices5
Vertex configuration3×(32)+6×(32)
Symmetry groupD3h
Dihedral angle (degrees)As a Johnson solid:Template:Bulletlist
Dual polyhedrontriangular prism
Propertiesconvex,
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]Template:R/superscript[2]Template:R/superscript or trigonal bipyramid.[3]Template:R/superscript 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.

Special cases

As a right bipyramid

Like other bipyramids, a triangular bipyramid can be constructed by attaching two tetrahedra face-to-face.[2]Template:R/superscript These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, five vertices, and nine edges.[3]Template:R/superscript Because of its triangular faces with any type, the triangular bipyramid is a simplicial polyhedron like other infinitely many bipyramids.[4]Template:R/superscript 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]Template:R/superscript If two tetrahedra are otherwise, the triangular bipyramid is oblique.[6]Template:R/superscript[7]Template:R/superscript

A line drawing with multicolored dots
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 (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]Template:R/superscript[9]Template:R/superscript

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]Template:R/superscript A triangular bipyramid is face-transitive (or isohedral).[11]Template:R/superscript

As a Johnson solid

Page Template:Multiple image/styles.css has no content.

A triangular bipyramid with regular faces
Multicolor, flat image of a triangular bipyramid
Triangular bipyramid with regular faces alongside its net
A grayscale image
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]Template:R/superscript 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]Template:R/superscript It is an example of a composite polyhedron because it is constructed by attaching two regular tetrahedra.[13]Template:R/superscript[14]Template:R/superscript

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]Template:R/superscript 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]Template:R/superscript

File:Kleetope of a triangular bipyramid.svg
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]Template:R/superscript[17]Template:R/superscript Another type of triangular bipyramid results from cutting off its vertices, a process known as truncation.[18]Template:R/superscript

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]Template:R/superscript[3]Template:R/superscript A triangular prism has five faces, nine edges, and six vertices, with the same symmetry as a triangular bipyramid.[3]Template:R/superscript

Applications

Four circles, with geometric figures inside them
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]Template:R/superscript This solution is aided by a mathematically rigorous computer.[21]Template:R/superscript

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]Template:R/superscript Examples of this structure include phosphorus pentafluoride and phosphorus pentachloride in the gaseous phase.[23]Template:R/superscript

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]Template:R/superscript[25]Template:R/superscript

References

  1. ^ a b Page Module:Citation/CS1/styles.css has no content.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.
  2. ^ a b Page Module:Citation/CS1/styles.css has no content.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.
  3. ^ a b c d Page Module:Citation/CS1/styles.css has no content.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.
  4. ^ Page Module:Citation/CS1/styles.css has no content.Kumar, C. P. Anil (2020). "On the Coherent Labelling Conjecture of a Polyhedron in Three Dimensions". arXiv:1801.08685 [math.CO].
  5. ^ Page Module:Citation/CS1/styles.css has no content.Montroll, John (2011). Origami Polyhedra Design. CRC Press. p. 6. ISBN 978-1-4398-7106-5..
  6. ^ Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.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. arXiv:1607.06604. doi:10.1007/s13366-017-0336-8.
  8. ^ Page Module:Citation/CS1/styles.css has no content.Tutte, W. T. (2001). Graph Theory. Cambridge University Press. p. 113. ISBN 978-0-521-79489-3.
  9. ^ Page Module:Citation/CS1/styles.css has no content.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. S2CID 261797042.
  10. ^ Page Module:Citation/CS1/styles.css has no content.Alexander, Daniel C.; Koeberlin, Geralyn M. (2014). Elementary Geometry for College Students (6th ed.). Cengage Learning. p. 403. ISBN 978-1-285-19569-8.
  11. ^ Page Module:Citation/CS1/styles.css has no content.McLean, K. Robin (1990). "Dungeons, dragons, and dice". The Mathematical Gazette. 74 (469): 243–256. doi:10.2307/3619822. JSTOR 3619822. S2CID 195047512.
  12. ^ Page Module:Citation/CS1/styles.css has no content.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. S2CID 220150682.
  13. ^ Page Module:Citation/CS1/styles.css has no content.Timofeenko, A. V. (2009). "Convex Polyhedra with Parquet Faces" (PDF). Doklady Mathematics. 80 (2): 720–723. doi:10.1134/S1064562409050238.
  14. ^ a b Page Module:Citation/CS1/styles.css has no content.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.
  15. ^ Page Module:Citation/CS1/styles.css has no content.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.
  16. ^ Page Module:Citation/CS1/styles.css has no content.Grünbaum, Branko (1967). Convex Polytopes. Wiley Interscience. p. 357.. Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, Template:ISBN.
  17. ^ Page Module:Citation/CS1/styles.css has no content.Ewald, Günter (1973). "Hamiltonian circuits in simplicial complexes". Geometriae Dedicata. 2 (1): 115–125. doi:10.1007/BF00149287. S2CID 122755203.
  18. ^ Page Module:Citation/CS1/styles.css has no content.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..
  19. ^ Page Module:Citation/CS1/styles.css has no content.Sibley, Thomas Q. (2015). Thinking Geometrically: A Survey of Geometries. Mathematical Association of America. p. 53. ISBN 978-1-939512-08-6.
  20. ^ Page Module:Citation/CS1/styles.css has no content.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
  21. ^ Page Module:Citation/CS1/styles.css has no content.Schwartz, Richard Evan (2013). "The Five-Electron Case of Thomson's Problem". Experimental Mathematics. 22 (2): 157–186. doi:10.1080/10586458.2013.766570. S2CID 38679186.
  22. ^ Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.Housecroft, C. E.; Sharpe, A. G. (2004). Inorganic Chemistry (2nd ed.). Prentice Hall. p. 407. ISBN 978-0-13-039913-7.
  24. ^ Page Module:Citation/CS1/styles.css has no content.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.
  25. ^ Page Module:Citation/CS1/styles.css has no content.Kuehni, Rolf G. (2013). Color: An Introduction to Practice and Principles. John & Sons Wiley. p. 198. ISBN 978-1-118-17384-8.

Template:Johnson solids navigator

Retrieved from "https://cosmopedia.co/index.php?title=Triangular_bipyramid&oldid=388399"