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 ....

, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid

Archimedean solid

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

 with eight triangular

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 and eighteen square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

 faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles while the other twelve share an edge. The polyhedron

Polyhedron

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

 has octahedral symmetry

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...

, like the cube and 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....

. Its dual

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...

 is called the deltoidal icositetrahedron

Deltoidal icositetrahedron

In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron....

 or trapezoidal icositetrahedron, although its faces are not really true trapezoid

Trapezoid

In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

s.

The name rhombicuboctahedron refers to the fact that 12 of the square faces lie in the same planes as the 12 faces of the rhombic dodecahedron

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

 which is dual to 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,...

.