Type: | Johnson |
Faces: | 8 triangles 4 squares |
Edges: | 20 |
Vertices: | 10 |
Symmetry: | D4h |
Vertex Config: | 2 x 34+8 x (32 x 42) |
Net: | Elongated square bipyramid (symmetric net).svg |
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.
A zircon crystal is an example of an elongated square bipyramid.
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.. 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
J15
Given that
a
a
(1/\sqrt{2})a
\arccos(-1/3) ≈ 109.47\circ
\pi/2
\arctan\left(\sqrt{2}\right) ≈ 54.74\circ
D4h
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::.
It can be considered a transitional phase between the cubic and rhombic dodecahedral honeycombs. 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]
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.)[1] This honeycomb can be considered an elongated version of the tetrahedral-octahedral honeycomb.