Gyroelongated square bipyramid

Short description: Convex polyhedron with 16 triangular faces

Gyroelongated square bipyramid

Type| Gyroelongated bipyramid,
Deltahedron,
Johnson
J16 – J17 – J18
Faces| 16 triangles
Edges| 24
Vertices| 10
Vertex configuration| 2×(34)+8×(35)
Symmetry group| D4d
Properties| convex
Net

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron[1], heccaidecadeltahedron,[2] or tetrakis square antiprism;[1] these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

The dual polyhedron of the gyroelongated square bipyramid is a truncated square trapezohedron with eight pentagons and two squares as its faces. The gyroelongated square pyramid appears in chemistry as the basis for the bicapped square antiprismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem.

## Contents

* 1 Construction
* 2 Properties
* 3 Application
* 4 References

## Construction

Like other gyroelongated bipyramids, the gyroelongated square bipyramid can be constructed by attaching two equilateral square pyramids onto the square faces of a square antiprism; this process is known as gyroelongation.[3][4] These pyramids cover each square, replacing it with four equilateral triangles, so that the resulting polyhedron has 16 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the gyroelongated square bipyramid.[5] More generally, the convex polyhedron in which all faces are regular is the Johnson solid, and every convex deltahedron is a Johnson solid. The gyroelongated square bipyramid is numbered among the Johnson solids as J17.[6]

One possible system of Cartesian coordinates for the vertices of a gyroelongated square bipyramid, giving it edge length 2, is:[1] (±1,±1,2−1/4),(±2,0,−2−1/4),(0,±2,−2−1/4),(0,0,±(2−1/4+2)).

## Properties

The surface area of a gyroelongated square bipyramid is 16 times the area of an equilateral triangle, that is:[4] 43a2≈6.928a2, and the volume of a gyroelongated square bipyramid is obtained by slicing it into two equilateral square pyramids and one square antiprism, and then adding their volume:[4] 2+4+323a3≈1.428a3.

File:J17 gyroelongated square bipyramid.stl It has the same three-dimensional symmetry group as the square antiprism, the dihedral group of D4d of order 8. Its dihedral angle is similar to the gyroelongated square pyramid, by calculating the sum of the equilateral square pyramid and the square antiprism's angle in the following:[7]

* the dihedral angle of an equilateral square pyramid between two adjacent triangles, approximately 109.47∘
* the dihedral angle of a square antiprism between two adjacent triangles, approximately 127.55∘
* the dihedral angle between two adjacent triangles, on the edge where an equilateral square pyramid is attached to a square antiprism, is 158.57∘, which is obtained by adding the dihedral angles between a square and a triangle of both the pyramid and the antiprism.

Square truncated trapezohedron

The dual polyhedron of a gyroleongated square bipyramid is the square truncated trapezohedron.[citation needed] As suggested by the name, it is the family of truncated trapezohedron by means of truncating two axial vertices of a square trapezohedron. The resulting polyhedron has eight pentagons and two squares.[8]

## Application

Gyroelongated square bipyramid can be visualized in the geometry of chemical compounds as the atom cluster surrounding a central atom as a polyhedron, and the compound of such cluster is the bicapped square antiprismatic molecular geometry.[9] It has 10 vertices and 24 edges, corresponding to the closo polyhedron with 2n+2 skeletal electrons. An example is nickel carbonyl carbide anion Ni10C(CO)2−18, a 22 skeletal electron chemical compound with ten Ni(CO)2 vertices and the deficiency of two carbon monoxides.[10]

The Thomson problem concerning the minimum-energy configuration of n charged particles on a sphere. The minimum solution known for n=10 places the points at the vertices of a gyroelongated square bipyramid, inscribed in a sphere.[1]

## References

1. ↑ 1.0 1.1 1.2 1.3 "Minimal-energy clusters of hard spheres", Discrete & Computational Geometry 14 (3): 237–259, 1995, doi:10.1007/BF02570704 .
2. ↑ Pugh, Anthony (1976), Polyhedron: A Visual Approach, University of California Press, p. 35 .
3. ↑ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4, https://books.google.com/books?id=afJdDwAAQBAJ .
4. ↑ 4.0 4.1 4.2 Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8 .
5. ↑ Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine 51 (1): 55-57, doi:10.2307/2689647 .
6. ↑ Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways 46 (3): 177, https://digitalcommons.butler.edu/wordways/vol46/iss3/9/ .
7. ↑ "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, 1966, doi:10.4153/cjm-1966-021-8 .
8. ↑ de Corato, Marzio; Prosperio, Davide M.; Bernasconi, Marco; Benedek, Giorgio (2013), "Two C28 Clathretes", Diamond and Related Nanostructures, 6, Springer, p. 80-81, doi:10.1007/978-94-007-6371-5, ISBN 978-94-007-6371-5, Bibcode: 2013CMCP....6.....D, https://books.google.com/books?id=qtEMeCWakpAC&pg=PA80 .
9. ↑ Remhov, Arndt; Černý, Radovan (2021), "Hydroborate as novel solid-state electrolytes", Crystallography in Materials Science: From Structure-Property Relationships to Engineering, de Gruyter, p. 270, ISBN 978-3-11-067485-9, https://books.google.com/books?id=Hpc9EAAAQBAJ&pg=PA270 .
10. ↑ King, R. Bruce (1993), Applications of Graph Theory and Topology in In Cluster and Coordination Chemistry, CRC Press, p. 102, ISBN 978-0-8493-4298-1, https://books.google.com/books?id=YA0ltXgmB54C&pg=PA102 .

* v
* t
* e

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)

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Gyroelongated square bipyramid. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Gyroelongated_square_bipyramid&oldid=4228871"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template