The Truncated Octahedron
The truncated octahedron is a 3D uniform polyhedron bounded by 14 polygons (8 hexagons and 6 squares), 36 edges, and 24 vertices. It may be constructed by truncating the octahedron at 1/3 of its edge length.
The dual of the truncated octahedron is the tetrakis hexahedron, a Catalan solid.
Projections
In order to be able to identify the truncated octahedron 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 

Octagon  Parallel projection centered on a square face. The top, bottom, left, and right edges of the projection envelope are images of square faces. 

Dodecagon  Parallel projection centered on a hexagonal face. 

Hexagon  Parallel projection centered on an edge between two hexagons. The top and bottom edges of the projection envelope are images of square faces. 

Nonuniform decagon  Parallel projection centered on a vertex. 
Coordinates
The Cartesian coordinates of the truncated octahedron, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of:
 (0, √2, 2√2)
Occurrences
The truncated octahedron appears as cells in the following 4D uniform polytopes:
 The cantitruncated 5cell;
 The omnitruncated 5cell;
 The bitruncated tesseract;
 The omnitruncated tesseract;
 The truncated 24cell;
 The runcitruncated 24cell;
 The cantitruncated 600cell;
 The omnitruncated 120cell.
It also occurs as cells in the following CRF polychora: