Elongated square bipyramid

Cube capped by two square pyramids

Elongated square bipyramid

Type| Johnson
J14 – J15 – J16
Faces| 8 triangles
4 squares
Edges| 20
Vertices| 10
Vertex configuration|  2 × 3 4 + 8 × ( 3 2 × 4 2 ) {\displaystyle 2\times 3^{4}+8\times (3^{2}\times 4^{2})}
Symmetry group|  D 4 h {\displaystyle D_{4h}}
Dual polyhedron| square bifrustum
Properties| convex
Net

In geometry, the elongated square bipyramid (or elongated octahedron) is the polyhedron constructed by attaching two equilateral square pyramids onto a cube's faces that are opposite each other. 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 is also been named the pencil cube or 12-faced pencil cube due to its shape.[1][2]

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

## Construction[edit]

The elongated square bipyramid is constructed by attaching two equilateral square pyramids onto the faces of a cube that are opposite each other, a process known as elongation. This construction involves the removal of those two squares and replacing them with those pyramids, resulting in eight equilateral triangles and four squares as their faces.[3]. A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as  J 15 {\displaystyle J_{15}} , the fifteenth Johnson solid.[4]

## Properties[edit]

Given that  a {\displaystyle a} is the edge length of an elongated square bipyramid. The height of an elongated square pyramid can be calculated by adding the height of two equilateral square pyramids and a cube. The height of a cube is the same as the given edge length  a {\displaystyle a} , and the height of an equilateral square pyramid is  ( 1 / 2 ) a {\displaystyle (1/{\sqrt {2}})a} . Therefore, the height of an elongated square bipyramid is:[5]

a + 2 ⋅ 1 2 a = ( 1 + 2 ) a ≈ 2.414 a . {\displaystyle a+2\cdot {\frac {1}{\sqrt {2}}}a=\left(1+{\sqrt {2}}\right)a\approx 2.414a.}

Its surface area can be calculated by adding all the area of eight equilateral triangles and four squares:[6]

( 4 + 2 3 ) a 2 ≈ 7.464 a 2 . {\displaystyle \left(4+2{\sqrt {3}}\right)a^{2}\approx 7.464a^{2}.}

Its volume is obtained by slicing it into two equilateral square pyramids and a cube, and then adding them:[6]

( 1 + 2 3 ) a 3 ≈ 1.471 a 3 . {\displaystyle \left(1+{\frac {\sqrt {2}}{3}}\right)a^{3}\approx 1.471a^{3}.}

Its dihedral angle can be obtained in a similar way as the elongated square pyramid, by adding the angle of square pyramids and a cube. In an equilateral square pyramid, the dihedral angle between square and triangle is  arctan ⁡ ( 2 ) ≈ 54.74 ∘ {\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }} , and that between two adjacent triangles is  arccos ⁡ ( − 1 / 3 ) ≈ 109.47 ∘ {\displaystyle \arccos(-1/3)\approx 109.47^{\circ }} . The dihedral angle between two adjacent squares in a cube is  π / 2 {\displaystyle \pi /2} . Therefore, the dihedral angle of the triangle and square, on the edge where the square pyramids attach the cube, is:[7]

arctan ⁡ ( 2 ) + π 2 ≈ 144.74 ∘ . {\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}

3D model of an elongated square bipyramid

The elongated square bipyramid has the dihedral symmetry, the dihedral group D 4 h {\displaystyle D_{4h}} of order eight: it has an axis of symmetry passing through the apices of square pyramids and the center of a cube, and its appearance is symmetrical by reflecting across a horizontal plane.[7]

## Related polyhedra and honeycombs[edit]

The elongated square bipyramid is dual to the square bifrustum, which has eight trapezoidals and two squares.

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.

The honeycomb

The half-honeycomb

It can be considered a transitional phase between the cubic and rhombic dodecahedral honeycombs.[1] Here, the cells are 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 ] {\displaystyle [4,3,4]} .

Chamfered square tiling

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

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.)[8] This honeycomb can be considered an elongated version of the tetrahedral-octahedral honeycomb.

## References[edit]

1. ^ a b Critchlow, Keith. Order in Space: A design source book. p. 46-47.
2. ^ Goldberg, Michael (January 1981). "On the space-filling octahedra". Geometriae Dedicata. 10 (1): 323–335. doi:10.1007/BF01447431.
3. ^ Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84-89\. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
4. ^ Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62\. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. S2CID 220150682.
5. ^ Sapiña, R. "Area and volume of the Johnson solid  J 15 {\displaystyle J_{15}} ". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-09.
6. ^ a b 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.
7. ^ 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. S2CID 122006114. Zbl 0132.14603.
8. ^ "J15 honeycomb".

## External links[edit]

* Weisstein, Eric W., "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
* 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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