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.
 A: 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:

A: I got here via A. Schulz's blog entry. As he says, you are looking for the projection of the $(d+1)$-dimensional hypercube (the $(d+1)$-cube for short) along a diagonal. But, actually, any generic projection of the $(d+1)$-cube will give (combinatorially) the same polytope. By "generic" I just mean that the direction of projection is not parallel to any coordinate hyperplane.
A different description is that your polytope is the zonotope obtained as the Minkowski sum of any set of $d+1$ generic vectors in $\mathbb R^d$, where generic means that no $d$ of them lie in a hyperplane.
A: Is there any chance that you are looking for the name of a {3,4,3} regular convex polychoron? It is called 24-cell (or icositetrachoron, octaplex, octacube polyoctahedron). 

24-cell can be derived as a rectified 16-cell (different from 16-cell).

Other related polychrons (uniform, but not necessary regular and convex):


Or if you are looking for a {5,3,3} regular and convex 4D polytope, it is called 120-cell (or hecatonicosachoron).
 

Here is a table I found on Wikipedia that describes the polytope families. Hope you find it useful.

A: The polytopes you describe are the duals of a family of uniform polytopes.
So, they are higher-dimensional analogues of the Catalan solids.
I have currently no better idea for how to describe these uniform polytope to you other than by their Coxeter diagrams:
$\qquad\qquad$
They belong to the $A_d$-family of uniform polytopes, that is, they have (an extended version of) the symmetry group of the $d$-simplex.
According to a link posted by A. Schulz, these polytopes are also called "cats-eye polytopes", which I think is pretty cool.
