CubeOctahedron, an Archimedean Solid

A CubeOctahedron is the intersection of a cube and an octahedron if these two are scaled so that their corresponding edges intersect at their midpoints. See the following animation.
The cubeoctahedron is obtained in the same way by midpoint truncation from the cube and from the octahedron.
The Archimedean solid that is often called "truncated cubeoctahedron" is not quite a truncation, because vertex truncation produces rectangles rather than squares. An additional slight shift of the vertices is required. Still, the animation looks like a truncation of the cubeoctahedron.
The cubeoctahedron inside its dual, the rhombic dodecahedron (wireframe).
The cubeoctahedron vertices touch the midpoints of the rhombic dodecahedron faces.
If one constructs the cubeoctahedron as the intersection of a cube and an octahedron whose edge midpoints have the same distance from the body midpoint, then the dual polyhedron has the vertices of the cube as 3-edged vertices and the it has the vertices of the octahedron as 4-edged vertices.