Projection of a hypercube along a long diagonal. What is this polytope called? I was wondering what people call a certain type of shape. It is the shape formed by an orthogonal projection of a hypercube along one of its longest diagonals.
In other words, fill in the missing entries:
n-simplex : Triangle ->      Tetrahedon        ->  5-cell    ...    
 n-cube   :  Square  ->         Cube           -> Tesseract  ...  
   ?      :  Hexagon -> Rhomibic Dodecahedron  ->     ?      ...

IMPORTANT: I'm looking for the name of the class of shapes, not just the 4D analogue.

What you are looking for is the vertex-frist projection of a hypercube. Apparently it seems that the 4d analogue has no special name.
You can read this post for more informations. 
Edit: I computed the polytope. Read my blog post about it. Here is a teaser:

