Vertex-first projections of hypercubes

A few days ago I was answering a question posted on math.stackexchange.com. It was asked what would be the next polytope in the following sequence "Hexagon,Rhombic Dodecahedron,???"?

One possible answer for this question goes along the following lines: Both the hexagon and the rhombic dodecahedron are vertex-first projections of cubes. The hexagon is the projection of the 3-cube, and the rhombic dodecahedron is the projection of the 4-cube. So the next polytope in this sequence would be the vertex-first projection of the 5d cube. (I consider the vertex-first projection that aligns two opposing cube vertices along the normal vector of the projection hyperplane.)

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