Tetrakis cuboctahedron
Type	Conway polyhedron Conway polyhedron notation Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prisms , antiprisms and pyramids...
Faces	32 triangles (2 types)
Edges	48 (2 types)
Vertices	18 (2 types)
Vertex configuration Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... s	(6) 35 (12) 36
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	Octahedral Octahedral symmetry 150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...  (Oh)
Dual polyhedron Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...	Truncated rhombic dodecahedron Truncated rhombic dodecahedron The truncated rhombic dodecahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices.The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares.The...
Properties	convex Convex set In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

The tetrakis cuboctahedron is a convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

 polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 with 32 triangular faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

, 48 edges

Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

, and 18 vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

. It a dual of the truncated rhombic dodecahedron

Truncated rhombic dodecahedron

The truncated rhombic dodecahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices.The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares.The...

.

Its name comes from a topological construction from the 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. As such it is a quasiregular polyhedron,...

 with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained the 6 order-4 vertices at a different distance from the center as the other 12.

Related polyhedra

It can also be topologically constructed from the octahedron

Octahedron

In geometry, 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 vertex....

, dividing each triangular face into 4 triangles by adding mid-edge vertices (An ortho operation. From this construction, all 80 triangles will be equilateral.

This polyhedron can be confused with a slightly smaller Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

, the tetrakis hexahedron

Tetrakis hexahedron

In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube....

, which has only 24 triangles, 32 edges, and 14 vertices.

External links

• VTML polyhedral generator Try "k4aC" (Conway polyhedron notation
  Conway polyhedron notation
  Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prisms , antiprisms and pyramids...
  
  )