The Cuboctahedron
The cuboctahedron is a uniform polyhedron bounded by 14 polygons (6 squares and 8 triangles), 24 edges, and 12 vertices. It is edgeuniform, and its two kinds of faces alternate around each vertex, so it is also a quasiregular polyhedron. It may be constructed by truncating a cube or an octahedron at the midpoints of its edges (this process is known as rectification).
The cuboctahedron can be bisected by a plane parallel to two opposite triangular faces to produce the triangular cupola (J3), a Johnson solid.
Projections
In order to be able to identify the cuboctahedron in various projections of 4D objects, it is useful to know how it appears from various viewpoints. The following are some of the commonlyencountered views:
Projection  Envelope  Description 

Square  Parallel projection centered on a square face. 

Regular hexagon  Parallel projection centered on a triangular face. 

Hexagon  Vertexfirst projection. 

Hexagon  Edgefirst projection. 
Coordinates
The Cartesian coordinates of the cuboctahedron, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of:
 (0, √2, √2)
Occurrences
The cuboctahedron occurs as cells in a variety of 4D uniform polytopes:
 The cantellated 5cell;
 The runcitruncated 5cell;
 The rectified tesseract;
 The runcitruncated tesseract;
 The rectified 24cell;
 The cantellated 24cell;
 The runcitruncated 120cell;
 The cantellated 600cell.
Some CRF polychora also contain cuboctahedral cells, including (but not limited to):