Elongated square bipyramid

Short description: 15th Johnson solid; cube capped by 2 square pyramids

Elongated square bipyramid

Type| Johnson
J14 – J15 – J16
Faces| 8 triangles
4 squares
Edges| 20
Vertices| 10
Vertex configuration| 2(34)
8(32.42)
Symmetry group| D4h, [4,2], (*422)
Rotation group| D4, [4,2]+, (422)
Dual polyhedron| Square bifrustum
Properties| convex
Net

File:Johnson J15 3D.stl

In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (J15). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves. It can also be seen as 4 lunes (squares with triangles on opposite sides) linked together with squares to squares and triangles to triangles.

It has been named the pencil cube or 12-faced pencil cube due to its shape.[1][2]

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[3]

A zircon crystal is an example of an elongated square bipyramid.

## Contents

* 1 Formulae
* 2 Dual polyhedron
* 3 Related polyhedra and honeycombs
* 4 See also
* 5 References
* 6 External links

## Formulae

The following formulae for volume ([math]\displaystyle{ V }[/math]), surface area ([math]\displaystyle{ A }[/math]) and height ([math]\displaystyle{ H }[/math]) can be used if all faces are regular, with edge length [math]\displaystyle{ L }[/math]:[4]

[math]\displaystyle{ V = L^3\cdot \left( 1 + \frac{\sqrt{2}}{3}\right) \approx L^3\cdot 1.471404521 }[/math]
[math]\displaystyle{ A = L^2\cdot \left(4 + 2\sqrt{3}\right) \approx L^2\cdot 7.464101615 }[/math]
[math]\displaystyle{ H = L\cdot \left( 1 + \sqrt{2}\right) \approx L\cdot 2.414213562 }[/math]

## Dual polyhedron

The dual of the elongated square bipyramid is called a square bifrustum and has 10 faces: 8 trapezoidal and 2 square.

Dual elongated square bipyramid  | Net of dual

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## Related polyhedra and honeycombs

A special kind of elongated square bipyramid without all regular faces allows a self-tessellation of Euclidean space. The triangles of this elongated square bipyramid are not regular; they have edges in the ratio 2:√3:√3.

It can be considered a transitional phase between the cubic and rhombic dodecahedral honeycombs.[1] The cells are here colored white, red, and blue based on their orientation in space. The square pyramid caps have shortened isosceles triangle faces, with six of these pyramids meeting together to form a cube. The dual of this honeycomb is composed of two kinds of octahedra (regular octahedra and triangular antiprisms), formed by superimposing octahedra into the cuboctahedra of the rectified cubic honeycomb. Both honeycombs have a symmetry of 4,3,4.

Cross-sections of the honeycomb, through cell centers produces a chamfered square tiling, with flattened horizontal and vertical hexagons, and squares on the perpendicular polyhedra.

Honeycomb  |
Half honeycomb  |
Chamfered square tiling

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With regular faces, the elongated square bipyramid can form a tessellation of space with tetrahedra and octahedra. (The octahedra can be further decomposed into square pyramids.)[5] This honeycomb can be considered an elongated version of the tetrahedral-octahedral honeycomb.

## See also

* Elongated square pyramid

## References

1. ↑ 1.0 1.1 Order in Space: A design source book, Keith Critchlow, p.46-47
2. ↑ Goldberg, Michael, On the space-filling octahedra, Geometriae Dedicata, January 1981, Volume 10, Issue 1, pp 323–335 [1] PDF
3. ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8 .
4. ↑ Sapiña, R.. "Area and volume of the Johnson solid J15" (in es). Problemas y Ecuaciones. ISSN 2659-9899. https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J15/calculadora-area-volumen-formulas.html. Retrieved 2020-09-09.
5. ↑ "J15 honeycomb". http://woodenpolyhedra.web.fc2.com/J15.html.

## External links

* Eric W. Weisstein, Elongated square bipyramid (Johnson solid) at MathWorld.

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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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