The Square Orthobicupola
The square orthobicupola is the 28th Johnson solid (J28). It has 16 vertices, 32 edges, and 18 faces (8 equilateral triangles and 10 squares).
The square orthobicupola can be constructed by joining two square cupolae to each other at their octagonal face, such that the triangular faces touch each other at the edges. The ortho in the name refers to how the orientation of the two cupolae line up with each other. Joining the two cupolae in gyro orientation produces the square gyrobicupola instead.
An octagonal prism can be inserted between the two cupolae, lengthening the square orthobicupola into a rhombicuboctahedron.
Projections
Here are some views of the square orthobicupola from various angles:
Projection  Description 

Top view. 

Front view. 

45° side view. 
Coordinates
The Cartesian coordinates of the square orthobicupola with edge length 2 are:
 (±1, ±1, ±√2)
 (±1, ±(1+√2), 0)
 (±(1+√2), ±1, 0)
Occurrences
The square orthobicupola occurs as cells in a number of CRF polytopes (generalizations of the Johnson solids to 4D), including, but not limited to: