The Gyroelongated Square Pyramid
The gyroelongated square pyramid is the 10th Johnson solid (J10). It has 9 vertices and 13 faces (12 equilateral triangles and 1 square).
The gyroelongated square pyramid can be constructed by attaching a square pyramid to a square antiprism.
Attaching a second square pyramid to the other side of the antiprism produces the gyroelongated square bipyramid (J17).
Projections
Here are some views of the gyroelongated square pyramid from various angles:
Projection  Envelope  Description 

Regular octagon  Top view. 

Pentagon  Side view. 

Pentagon  Diagonal side view. 

Pentagon  22.5° side view. 
Coordinates
The Cartesian coordinates of the gyroelongated square pyramid with edge length 2 are:
 (0, 0, √2 + 1/∜2)
 (0, ±√2, 1/∜2)
 (±√2, 0, 1/∜2)
 (±1, ±1, −1/∜2)