The Octagonal Antiprism
The octagonal antiprism is a 3D uniform polyhedron bounded by 2 octagons and 16 triangles. It has 32 edges and 16 vertices.
The height of an octagonal antiprism with an edge length of 2 is:
2√((1 + √2)(√(1 + 1/√2)  1))
which is approximately 1.720591.
The square cupola (J4) can be pasted with an octagonal antiprism to produce a gyroelongated square cupola (J23). Adding a second cupola produces the gyroelongated square bicupola (J45).
Projections
Here are some views of the octagonal antiprism from various angles:
Projection  Envelope  Description 

Regular hexadecagon  Parallel projection centered on octagonal face. 

Trapezium  Parallel projection parallel to square faces and a pair of triangles. 

Rectangle  11.25° side view. 

Nonagon  Parallel projection centered on vertex. 
Coordinates
The Cartesian coordinates of the octagonal antiprism, centered on the origin and having edge length 2, are:
 (±1, ±(1+√2), H)
 (±(1+√2), ±1, H)
 (0, ±√(4+2√2), H)
 (±√(4+2√2), 0, H)
 (±√(2+√2), ±√(2+√2), H)
where H = √((1 + √2)(√(1 + 1/√2)  1)), or approximately 0.860296, is half the height of the antiprism.