Elongated square bipyramid

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Elongated square bipyramid
Elongated square dipyramid.png
Type Johnson
J14 - J15 - J16
Faces 8 triangles
4 squares
Edges 20
Vertices 10
Vertex configuration 2(34)
Symmetry group D4h, [4,2], (*422)
Rotation group D4, [4,2]+, (422)
Dual polyhedron Square bifrustum
Properties convex
Johnson solid 15 net.png

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 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 have regular faces but are not uniform (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.

Dual polyhedron[edit]

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
Dual elongated square dipyramid.png Dual elongated square dipyramid net.png

Related polyhedra and honeycombs[edit]

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.

Elongated oblate octahedron.pngElongated oblate octahedron net.png

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.

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

Elongated oblate octahedron-red-green-blue-honeycomb.png
Half honeycomb
Elongated square tiling.svg
Chamfered square tiling

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

See also[edit]


  1. ^ a b 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, MR 0185507, Zbl 0132.14603, doi:10.4153/cjm-1966-021-8 .
  4. ^ http://woodenpolyhedra.web.fc2.com/J15.html

External links[edit]