Elongated square bipyramid

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

${\displaystyle V=L^{3}\cdot \left(1+{\frac {\sqrt {2}}{3}}\right)\approx L^{3}\cdot 1.471404521}$

${\displaystyle A=L^{2}\cdot \left(4+2{\sqrt {3}}\right)\approx L^{2}\cdot 7.464101615}$

${\displaystyle H=L\cdot \left(1+{\sqrt {2}}\right)\approx L\cdot 2.414213562}$

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

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

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.

• Elongated square pyramid

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