Pentagonal bipyramid

Pentagonal bipyramid
Type	Bipyramid, Deltahedra Johnson J12 – J13 – J14 Simplicial
Faces	10 triangles
Edges	15
Vertices	7
Vertex configuration	${\displaystyle 5\times (3^{4})+2\times (3^{5})}$
Symmetry group	${\displaystyle D_{5\mathrm {h} }}$
Dihedral angle (degrees)	As a Johnson solid: triangle-to-triangle: 138.2° triangle-to-triangle if pyramids being attached base-to-base: 74.8°
Dual polyhedron	pentagonal prism
Properties	convex, composite, face-transitive

The pentagonal bipyramid or pentagonal dipyramid ^[1] is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, composite polyhedron, and Johnson solid.

The pentagonal bipyramid may be represented as well-covered graph, which is four-connected. This polyhedron may be used in the chemical compound as the description of an atom cluster known as pentagonal bipyramidal molecular geometry, as a solution in Thomson problem, as well as in decahedral nanoparticles.

Special cases

As a right bipyramid

The pentagonal bipyramid can be constructed by attaching the bases of two pentagonal pyramids. ^[2] These pyramids cover their pentagonal base, such that the resulting polyhedron has ten triangles as its faces, fifteen edges, and seven vertices. ^[3] Because of its triangular faces with any type, the pentagonal bipyramid is a simplicial polyhedron like other infinitely many bipyramids. ^[4] The pentagonal bipyramid is said to be right if two pyramids are identical, meaning that they are symmetrically regular and their apices are on the line passing through the base's center, ^[5] and like any right bipyramid, its faces are isosceles triangles. ^[6] Two pyramids that are otherwise result in an oblique form. ^[5]

The right pentagonal bipyramid is face-transitive or isohedral, meaning any mapping of two adjacent faces preserves its symmetrical appearance by either the transformations of translations, rotations, or reflections. ^[7] This relates to the fact that it has three-dimensional symmetry group of dihedral group ${\displaystyle D_{5\mathrm {h} }}$ of order twenty: having rotation of one- up to four-fifth around the axis of symmetry that passing through apices and base's center vertically, mirror symmetry relative to any bisector of the base, and reflection it across a horizontal plane. ^[8]

As a Johnson solid

Pentagonal bipyramid with regular faces, alongside its net.

3D model of a pentagonal bipyramid as a Johnson solid

The pentagonal bipyramid is one of the eight convex deltahedra if the faces of two pyramids are equilateral triangles and all edges are of equal length. ^[1] It is an example of a composite polyhedron because it can be constructed by attaching two regular-faced pentagonal pyramids. ^{[9] [3]} Because of its faces are regular polygonal, such a pentagonal bipyramid is generally a Johnson solid; every convex deltahedron is a Johnson solid. It is designated as the thirteenth Johnson solid ${\displaystyle J_{13}}$ out of 92. ^[10]

A pentagonal bipyramid's surface area ${\displaystyle A}$ is 10 times that of all triangles, and its volume ${\displaystyle V}$ can be ascertained by slicing it into two pentagonal pyramids and adding their volume. In the case of edge length ${\displaystyle a}$, the formulations are: ^[3] $${\displaystyle A={\frac {5{\sqrt {3}}}{2}}a^{2}\approx 4.3301a^{2},\qquad V={\frac {5+{\sqrt {5}}}{12}}a^{3}\approx 0.603a^{3}.}$$

The dihedral angle of a regular-faced pentagonal bipyramid can be calculated by adding the angle of pentagonal pyramids: ^[11]

• the dihedral angle of a pentagonal bipyramid between two adjacent triangles is that of a pentagonal pyramid, approximately 138.2°, and
• the dihedral angle of a pentagonal bipyramid with regular faces between two adjacent triangular faces, on the edge where two pyramids are attached, is 74.8°, obtained by summing the dihedral angle of a pentagonal pyramid between the triangular face and the base.

The pentagonal bipyramid has one type of closed geodesic, the path on the surface avoiding the vertices and locally looks like the shortest path. In other words, this path follows straight line segments across each face that intersect, creating complementary angles on the two incident faces of the edge as they cross. The closed geodesic crosses the apical and equator edges of a pentagonal bipyramid, with the length of ${\displaystyle 2{\sqrt {3}}\approx 3.46}$. ^[12]

Graph

Graph of a pentagonal bipyramid. In the illustration, the yellow vertices are the apices of a bipyramid.

The vertices and edges of a pentagonal bipyramid can give rise to a graph. This is due to Steinitz's theorem stating that the edges of any convex polyhedron can have a planar graph which is 3-connected.^{[ citation needed ]} Being planar means that the edges cannot cross each other, and being ${\displaystyle k}$-connected means that the graph remains connected whenever ${\displaystyle k-1}$ vertices are removed. Similar to the regular octahedron, the snub disphenoid, and an irregular polyhedron with twelve vertices and twenty triangular faces, the graph of a pentagonal bipyramid is 4-connected simplicial well-covered, meaning that all of the maximal independent sets of its vertices have the same size (i.e., the same number of edges). ^[13]

The graph of a pentagonal bipyramid is one of only six connected graphs in which the neighborhood of every vertex is a cycle of length four or five; the others being the Fritsch graph, the octahedral graph, the icosahedral graph, and the edge graphs of the snub disphenoid and the gyroelongated square bipyramid. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a topological surface called a Whitney triangulation. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive angular defect at every vertex. This makes them a combinatorial analogue of the positively curved smooth surfaces. They come from six of the eight deltahedra—excluding the two that have a vertex with a triangular neighborhood. ^[14]

Related polyhedra

The dual polyhedron of a pentagonal bipyramid is the pentagonal prism. More generally, every bipyramid is dual to a prism. ^[15] The pentagonal prism has two pentagonal faces at the base, each of which connects its vertices with an edge, producing five more rectangular faces. ^[16]

A preprint by Gallet et al. (2024) claimed that there is another simpler non-self-crossing flexible polyhedron with only eight vertices, despite that Steffen's polyhedron with nine vertices has been claimed to be the simplest; being flexible means a polyhedron can be continuously changed while preserving the shape of its faces. They obtained it by combining two Bricard octahedra to form a self-crossing flexible pentagonal bipyramid, and then replacing one of its faces by three triangles to eliminate the self-crossing. ^[17]

Applications

In the geometry of chemical compounds, the pentagonal bipyramid can be used as the atom cluster surrounding an atom. The pentagonal bipyramidal molecular geometry describes clusters for which this polyhedron is a pentagonal bipyramid. An example of such a cluster is iodine heptafluoride in the gas phase. ^[18]

The pentagonal bipyramid with equilateral triangular faces is one of the known solutions of a seven-electron case in Thomson problem, concerning the minimum-energy configuration of charged particles on a sphere. The solution is done by placing the vertices of a pentagonal bipyramid inscribed in a sphere. ^[19]

Pentagonal bipyramids and related five-fold shapes are found in decahedral nanoparticles, which can also be macroscopic in size when they are also called fiveling cyclic twins in mineralogy. ^{[20] [21]}

References

1. 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. 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. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, Bibcode:1971FrInJ.291..329B, doi:10.1016/0016-0032(71)90071-8, MR   0290245 .
4. Kumar, C. P. Anil (2020), On the Coherent Labelling Conjecture of a Polyhedron in Three Dimensions, arXiv: 1801.08685 .
5. Polya, G. (1954), Mathematics and Plausible Reasoning: Induction and analogy in mathematics, Princeton University Press, p. 138, ISBN   0-691-02509-6 .
6. Montroll (2011), p.  6.
7. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR   3619822, S2CID   195047512 .
8. Alexander, Daniel C.; Koeberlin, Geralyn M. (2014). Elementary Geometry for College Students (6th ed.). Cengage Learning. p. 403. ISBN   978-1-285-19569-8.
9. Timofeenko, A. V. (2009), "Convex Polyhedra with Parquet Faces" (PDF), Doklady Mathematics, 80 (2): 720–723, doi:10.1134/S1064562409050238 .
10. 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 .
11. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics , 18: 169–200, Bibcode:1966CJMat..18..169J, doi: 10.4153/cjm-1966-021-8 , MR   0185507, S2CID   122006114, Zbl   0132.14603 .
12. Lawson, Kyle A.; Parish, James L.; Traub, Cynthia M.; Weyhaupt, Adam G. (2013), "Coloring graphs to classify simple closed geodesics on convex deltahedra." (PDF), International Journal of Pure and Applied Mathematics, 89 (2): 123–139, doi: 10.12732/ijpam.v89i2.1 , Zbl   1286.05048 .
13. Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010), "On well-covered triangulations. III", Discrete Applied Mathematics, 158 (8): 894–912, doi: 10.1016/j.dam.2009.08.002 , MR   2602814 .
14. Knill, Oliver (2019), A simple sphere theorem for graphs, arXiv: 1910.02708 .
15. Montroll, John (2011), Origami Polyhedra Design, CRC Press, p. 5, ISBN   978-1-4398-7106-5 .
16. Goldberg, Nicholas; Haylett, Christine (2019), Oxford Mathematics for the Caribbean: Book 1 (6th ed.), Oxford University Press, p. 308, ISBN   978-0-19-842569-4 .
17. Gallet, Matteo; Grasegger, Georg; Legerský, Jan; Schicho, Josef (October 17, 2024), Pentagonal bipyramids lead to the smallest flexible embedded polyhedron, arXiv: 2410.13811 .
18. Gillespie, Ronald J.; Hargittai, István (2013), The VSEPR Model of Molecular Geometry, Dover Publications, p. 152, ISBN   978-0-486-48615-4 .
19. 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 .
20. Marks, L. D.; Peng, L. (2016), "Nanoparticle shape, thermodynamics and kinetics", Journal of Physics: Condensed Matter, 28 (5) 053001, Bibcode:2016JPCM...28e3001M, doi: 10.1088/0953-8984/28/5/053001 , ISSN   0953-8984, PMID   26792459 .
21. Rose, Gustav (1831), "Ueber die Krystallformen des Goldes und des Silbers", Annalen der Physik, 99 (10): 196–204, Bibcode:1831AnP....99..196R, doi:10.1002/andp.18310991003, ISSN   0003-3804 .

External links

• Weisstein, Eric W., "Pentagonal dipyramid" ("Dipyramid") at MathWorld .
• Conway Notation for Polyhedra Try: dP5