# 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.

• It seems to me your arrows should point in the opposite direction---from right to left. Commented Dec 10, 2012 at 3:53
• @MichaelHardy The arrows represent the well known coolness ordering of polytopes, read "X -> Y" as "Y is cooler than X". I think I get your intuition, there are increasing freedoms the way I drew them. e.g. the tetrahedron uniquely specifies the triangle, but not vice-versa. Commented Dec 10, 2012 at 4:16

What you are looking for is the vertex-frist projection of a hypercube. Apparently it seems that the 4d analogue has no special name.

Edit: I computed the polytope. Read my blog post about it. Here is a teaser:

• Unfortunately, the link to the blog post no longer works. Commented Dec 20, 2020 at 15:13

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.

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.

• That isn't what I'm asking, sorry :( Possibly it is the dual (in some sense) of the 24-cell. Commented Dec 18, 2012 at 11:55
• Also, I want the name of a whole class. Commented Dec 18, 2012 at 11:55

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.