looking for finding the shape of a 3D polytope in $\mathbb{R^{6}}$

Question

does any body have information about below polytope?

Suppose we have 6 points in $\mathbb{R^{6}}$:

$(1,1,1,0,0,0),(0,1,1,1,0,0),(0,0,1,1,1,0),(0,0,0,1,1,1),(1,0,0,0,1,1),(1,1,0,0,0,1)$

what is convex-hull of these points?

I have these information:

1-Since the vertices are not affinely independent it is not simplex.

2-this is a 3-dimension polytope.

I need more information about the geometry of this polytope! number of facets? and more important is it a famous 3D shape?

Answer 1

Just calculate the distances between adjacent points. Those would be $\sqrt{2}$.
Next calculate the distances between one but adjacent ones. Those will be $\sqrt{4}=2$.
Finally calculate the distances between opposites (wrt. the given cycle). Those are $\sqrt{6}$.

That is, these length are in ratio $1:\sqrt{2}:\sqrt{3}$, which are well-known distances within a cube. In fact, you just described a non-planar hexagon, the Petrie polygon of the cube.

Alternatively, you could consider the hull of that Petrie polygon. Then this would be a narrow trigonal antiprism. The lacing triangles of which are diagonally divided squares. I.e. the large regular base triangles have sides with a factor $\sqrt{2}$ larger than the lacing edges.

You even could consider this figure as being a cube, when placed axially along its body diagonal, and having chopped off both its polar segments (pyramids).

--- rk

Answer 2

More general, when starting with a point half $0$ and half $1$ in $\mathbb{R}^{2n}$ together with its cyclical permutations, you would get according distances $\sqrt{2k}$, which are exactly the distances of a hypercube of $\mathbb{R}^{n}$ with edge size $\sqrt{2}$. Thus those given $2n$ points are contained in the vertex set of the $n$-dimensional hypercube. In fact, those define the non-planar Petrie polygon thereof.

Wrt. those Petrie polygons of general hypercubes cf. also https://en.wikipedia.org/wiki/Petrie_polygon#The_Petrie_polygon_projections_of_regular_and_uniform_polytopes . There you also find projections of the edge skeletons of those hypercubes in such an orientation each, that those Petrie polygons become planar regular polygons each.

--- rk