Tetrahemihexahedron

The tetrahemihexahedron is quite a special polyhedron.
It has an Euler characteristic of 1 so it has the some topology of a projective plane (so non orientable).
It's quit close to the Roman Surface (a 3D mapping of the projective plane).

It has 7 faces and is the only odd uniform polyhedron (apart of prisms).
It has 4 triangular faces (positioned on an octahedron) and 3 squares (virtualy) intersecting in the center.

Just like an projective plane, it is easier to visualize this object in 3D when double covering it (each part is shown twice). The double cover of the projective plane is the sphere and the double cover of the Tetrahemihexahedron  is the Cuboctahedron (8 triangles and 6 squares).

The tetrahemihexahedron is a projective polyhedron, and the only uniform projective polyhedron that immerses in Euclidean 3-space.

In this model, to avoid the intersection of the square planes in the 3D representation, the faces are not entirely  presented.
As a side effect, it is easier to visualize mobius strips embedded in this representation of the projective plane (punctured projective plane = mobius strip).

2 versions are presented  :
- only the internal parts of the faces (stairway version)
- faces with boders