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Truncated cuboctahedron
 
 
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The truncated cuboctahedron is 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 ....
. It has 12 square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 faces, 8 regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces, 6 regular octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
al faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
al symmetry), the truncated cuboctahedron is a zonohedron
Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
.

rnate interchangeable names are:

The name truncated cuboctahedron, given originally by Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
, is a little misleading.






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The truncated cuboctahedron is 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 ....
. It has 12 square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 faces, 8 regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces, 6 regular octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
al faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
al symmetry), the truncated cuboctahedron is a zonohedron
Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
.

Other names

Alternate interchangeable names are:
  • Rhombitruncated cuboctahedron
  • Great rhombicuboctahedron
  • Omnitruncated cuboctahedron


The name truncated cuboctahedron, given originally by Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
, is a little misleading. If you truncate
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
 a cuboctahedron
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....
 by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
s. However, the resulting figure is topologically
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.

The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of 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....
 which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.

One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See uniform great rhombicuboctahedron
Uniform great rhombicuboctahedron

In geometry, the Uniform great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17.It shares the vertex arrangement with the convex truncated cube....
.

Area and volume

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

Vertices

To derive the number of vertices, we note that each vertex is the meeting point of a square, hexagon, and octagon.

  • Each of the 12 squares with their 4 vertices contribute 48 vertices because .
  • Each of the 8 hexagons with their 6 vertices contribute 48 vertices because .
  • Each of the 6 octagons with their 8 vertices contribute 48 vertices because .


Therefore, there may seem to exist vertices. However, we have over-counted the vertices thrice since a square, hexagon, and octagon meet at each vertex. Consequently, we divide 144 by 3 to correct for our over-counting: .

Cartesian coordinates

The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all permutations of:
(±1, ±(1+v2), ±(1+v8))


See also

  • cube
  • cuboctahedron
    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....
  • 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....
  • truncated icosidodecahedron
    Truncated icosidodecahedron

    The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....

External links

  • The Encyclopedia of Polyhedra