geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the elongated square bipyramid (or elongated octahedron) is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s (). As the name suggests, it can be constructed by elongating an

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

by inserting a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

between its congruent halves. It has been named the pencil cube or 12-faced pencil cube due to its shape.Order in Space: A design source book, Keith Critchlow, p.46-47 A

zircon Zircon () is a mineral belonging to the group of nesosilicates and is a source of the metal zirconium. Its chemical name is zirconium(IV) silicate, and its corresponding chemical formula is Zr SiO4. An empirical formula showing some of the r ...

crystal is an example of an elongated square bipyramid.

Formulae

The following

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

e for

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

(

V

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

(

A

) and

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...

(

H

) can be used if all

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

are regular, with edge length

L

: :

V = L^3\cdot  \left( 1 + \frac\right) \approx L^3\cdot 1.471404521

A = L^2\cdot  \left(4 + 2\sqrt\right) \approx L^2\cdot 7.464101615

H = L\cdot  \left( 1 + \sqrt\right) \approx L\cdot 2.414213562

Dual polyhedron

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

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::. : Elongated oblate octahedron

It can be considered a transitional phase between the cubic and

rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal s ...

s. The cells are here colored white, red, and blue based on their orientation in space. The

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral 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 The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a r ...

. Both honeycombs have a symmetry of 4,3,4. Cross-sections of the honeycomb, through cell centers produces a

chamfered square tiling In geometry, the chamfered square tiling or semitruncated square tiling is a tiling of the Euclidean plane. It is a square tiling with each edge Chamfer (geometry), chamfered into new hexagonal faces. It can also be seen as the intersection of tw ...

, 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 In geometry, a honeycomb is a ''space filling'' or ''close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dime ...

with

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

and

octahedra In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

. (The octahedra can be further decomposed into

s.) This honeycomb can be considered an elongated version of the

tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names incl ...

References

External links

* Johnson solids Pyramids and bipyramids {{Polyhedron-stub

Formulae

Dual polyhedron

Related polyhedra and honeycombs

See also

References

External links