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 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.