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Cuboctahedron
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a cuboctahedron is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron
Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
, i.e. an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
, being vertex-transitive
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
 and edge-transitive.

Its dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 is the rhombic dodecahedron
Rhombic dodecahedron

The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
.

area A and the volume V of the cuboctahedron of edge length a are:

Cartesian coordinates for the vertices of a cuboctahedron (of edge length v2) centered at the origin are:
(±1,±1,0)
(±1,0,±1)
(0,±1,±1)


Geometric relations
A cuboctahedron can be obtained by taking an appropriate cross section
Cross section (geometry)

In geometry, a cross-section is the intersection of a body in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc....
 of a four-dimensional cross-polytope
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
.

A cuboctahedron has octahedral symmetry.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a cuboctahedron is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron
Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
, i.e. an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
, being vertex-transitive
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
 and edge-transitive.

Its dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 is the rhombic dodecahedron
Rhombic dodecahedron

The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
.

Other Names

  • Heptaparallelohedron (Buckminster Fuller
    Buckminster Fuller

    Richard Buckminster ?Bucky? Fuller was an American architect, author, designer, futurist, inventor, and visionary. He was the second president of Mensa International....
    )
    • Fuller applied the name "Dymaxion
      Dymaxion

      The word Dymaxion is a brand name that Buckminster Fuller used for several of his inventions. It is a portmanteau of "dynamic maximum tension" , however it has also been reported that the name is a combination of the words dynamic, maximum, and ion, per the ....
      " and Vector Equilibrium to this shape, used in an early version of the Dymaxion map
      Dymaxion map

      The Dymaxion map or Fuller map is a map projection of a World map onto the surface of a polyhedron, which can then be unfolded to a net in many different ways and flattened to form a two-dimensional map which retains most of the relative proportional integrity of the globe map....
      .
  • Rectified
    Rectification (geometry)

    In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
     cube     or rectified octahedron (Norman Johnson)
    • Also cantellated
      Cantellation (geometry)

      In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex....
       tetrahedron       by a lower symmetry.
  • Triangular gyrobicupola (analog to Triangular orthobicupola
    Triangular orthobicupola

    In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases....
    )


Area and volume

The area A and the volume V of the cuboctahedron of edge length a are:

Cartesian coordinates

The Cartesian coordinates for the vertices of a cuboctahedron (of edge length v2) centered at the origin are:
(±1,±1,0)
(±1,0,±1)
(0,±1,±1)


Geometric relations


A cuboctahedron can be obtained by taking an appropriate cross section
Cross section (geometry)

In geometry, a cross-section is the intersection of a body in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc....
 of a four-dimensional cross-polytope
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
.

A cuboctahedron has octahedral symmetry. Its first stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
 is the compound
Polyhedral compound

A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of star polygon#Star figuress such as the hexagram....
 of a cube and its dual octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, with the vertices of the cuboctahedron located at the midpoints of the edges of either.

The cuboctahedron is a rectified
Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
 cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 and also a rectified octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
.

It is also a cantellated
Cantellation (geometry)

In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex....
 tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
. With this construction it is given the Wythoff Symbol: 3 3 | 2.
Cantellated Tetrahedron
A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 and its vertices are equivalent under that group.

The edges of a cuboctahedron form four regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola
Triangular cupola

In geometry, the triangular cupola is one of the Johnson solids . It can be seen as half a cuboctahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....
, one of the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s; the cuboctahedron itself thus can also be called a triangular gyrobicupola
Bicupola (geometry)

In geometry, a bicupola is a solid formed by connecting two cupola on their bases.There are two classes of bicupola because each cupola half is bordered by alternating triangles and squares....
, the simplest of a series (other than the gyrobifastigium
Gyrobifastigium

In geometry, the gyrobifastigium is the 26th Johnson solid . It can be constructed by joining two face-regular triangular prism s along corresponding square faces, giving a half-turn to one prism....
 or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola
Triangular orthobicupola

In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases....
.

Both triangular bicupolae are important in sphere packing
Sphere packing

In mathematics, sphere packing problems concern arrangements of non-overlapping identical spheres which fill a space. Usually the space involved is three-dimensional Euclidean space....
. The distance from the solid's centre to its vertices is equal to its edge length. Each central sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's centre.

Cuboctahedra appear as cells in three of the convex uniform honeycomb
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
s and in nine of the convex uniform polychora
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
.

The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.

Related polyhedra


Uniform Polyhedron 43 T0

Cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
Uniform Polyhedron 43 T01

Truncated cube
Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....
Uniform Polyhedron 43 T1

cuboctahedron
Uniform Polyhedron 43 T12

Truncated octahedron
Truncated octahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
Uniform Polyhedron 43 T2

Octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....


Related polytopes


The cuboctahedron can be decomposed into a regular octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
 and eight irregular but equal octahedra in the shape of the convex hull of a cube with two opposite vertices removed. This decomposition of the cuboctahedron corresponds with the cell-first parallel projection of the 24-cell
24-cell

In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
 into 3 dimensions. Under this projection, the cuboctahedron forms the projection envelope, which can be decomposed into 6 square faces, a regular octahedron, and 8 irregular octahedra. These elements correspond with the images of 6 of the octahedral cells in the 24-cell, the nearest and farthest cells from the 4D viewpoint, and the remaining 8 pairs of cells, respectively.

Cultural occurrences

  • In the Star Trek
    Star Trek

    Star Trek is an American Science fiction on television entertainment series and media franchise. The Star Trek fictional universe created by Gene Roddenberry is the setting of six television series including the original 1966 Star Trek: The Original Series, in addition to ten feature films with Star Trek to be released on May 8,...
     episode By Any Other Name, crew members were changed into cuboctahedra
    Cuboctahedron

    In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
    .


See also

  • Cube
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
  • Icosidodecahedron
    Icosidodecahedron

    An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
  • Octahedron
    Octahedron

    An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
  • Rhombicuboctahedron
    Rhombicuboctahedron

    The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....
  • Truncated cuboctahedron
    Truncated cuboctahedron

    The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges....


External links

  • The Encyclopedia of Polyhedra