Gyroelongated square bipyramid

17th Johnson solid

Gyroelongated square bipyramid

Type| Gyroelongated bipyramid,
Deltahedron,
Johnson
J16 – J17 – J18
Faces| 16 triangles
Edges| 24
Vertices| 10
Vertex configuration|  2 × ( 3 4 ) + 8 × ( 3 5 ) {\displaystyle 2\times (3^{4})+8\times (3^{5})}
Symmetry group|  D 4 d {\displaystyle D_{4d}}
Dual polyhedron| Truncated square trapezohedron
Properties| convex
Net

In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron,[1] or tetrakis square antiprism is a polyhedron with 16 triangular faces, constructed by attaching two equilateral square pyramids onto the face of a square antiprism. All of its triangular faces are equilateral, making it one of the eight convex deltahedra.

The dual of the gyroelongated square bipyramid is a square truncated trapezohedron with 10 faces: 8 pentagons and 2 squares.

## Construction[edit]

Like other gyroelongated bipyramids, the gyroelongated square bipyramid is constructed by attaching two equilateral square pyramids onto the square faces of a square antiprism; this process is known as gyroelongation.[2][3] As a result, 16 faces are equilateral triangles, and the gyroelongated square bipyramid is deltahedron.[4] More generally, the Johnson solid is the convex polyhedron in which all faces are regular, which includes the deltahedra as well. That is, the gyroelongated square bipyramid is the Johnson solid enumerated as  J 17 {\displaystyle J_{17}} among them.[5]

## Properties[edit]

The surface area of a gyroelongated square bipyramid is 16 times the area of an equilateral triangle, that is:[3]

4 3 a 2 ≈ 6.928 a 2 , {\displaystyle 4{\sqrt {3}}a^{2}\approx 6.928a^{2},}

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:[3]

2 + 4 + 3 2 3 a 3 ≈ 1.428 a 3 . {\displaystyle {\frac {{\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}}{3}}a^{3}\approx 1.428a^{3}.}

3D model of a gyroelongated square bipyramid

It has the same three-dimensional symmetry group as the square antiprism, the dihedral group of  D 4 d {\displaystyle D_{4d}} 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: the dihedral angle of two adjacent triangles in an equilateral square pyramid is  109.47 ∘ {\displaystyle 109.47^{\circ }} , and that of square and triangle in the same pyramid is  54.74 ∘ {\displaystyle 54.74^{\circ }} . The dihedral angle of a square antiprism between the two adjacent triangles is  127.55 ∘ {\displaystyle 127.55^{\circ }} , and that between square and triangle is  103.83 ∘ {\displaystyle 103.83^{\circ }} . The dihedral angle of the square and triangle, on the edge where the square pyramid attaches the antiprism, is  103.83 ∘ + 54.74 ∘ = 158.57 ∘ {\displaystyle 103.83^{\circ }+54.74^{\circ }=158.57^{\circ }} .[5]

## See also[edit]

* Gyroelongated bipyramid
* Gyroelongated square pyramid

## References[edit]

1. ^ Pugh, Anthony (1976), Polyhedron: A Visual Approach, University of California Press, p. 35.
2. ^ 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.
3. ^ a b c 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.
4. ^ Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.2307/2689647, JSTOR 2689647.
5. ^ 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, Zbl 0132.14603.

## External links[edit]

* Weisstein, Eric W., "Gyroelongated Square Dipyramid" ("Johnson solid") at MathWorld.

* 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
* 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)

This polyhedron-related article is a stub. You can help Wikipedia by expanding it.

|

* v
* t
* e

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template