Given a convex polytope $P\subseteq\Bbb R^d$. I am going to construct a new polytope from its edges (I call it the edge polytope) with the following steps:
- Take the 1-skeleton of $P$.
- Extract the different edge directions $v_1,...,v_n\in\Bbb R^d$ from the skeleton. The edge directions should be normalized, i.e. $\|v_i\|=1$.
- Take the convex hull $P_e$ of the points $\pm v_i$. Then $P_e$ is the edge polytope of $P$.
Question: Is this construction known in the literature? If yes, what is its name and where to read about it?
On top of that, I am interested in applying the same construction to embedded graphs.
Example. Here are some polytopes together with their edge polytope:
- Regular polygons give regular polygons again, where a $2n$-gon gives the same $2n$-gone again (maybe rotated), and $(2n+1)$-gons give $(4n+2)$-gons.
- The cube (and hypercube of any dimension) gives the octahedron (or cross-polytope of the respective dimension).
- Tetrahedron, octahedron and cuboctahedron all give the cuboctahedron.
- Icosahedron, dodecahedron and icosidodecahedron all give the icosidodecahedron.
- Prisms give bi-pyramids.
It seems that the edge polytope will have the same symmetries as the original (or more, if the original was not centrally symmetric with all vertices on a sphere).
Further, it seems that iterating this construction ends in fixed points, e.g. $2n$-gons, the cuboctahedron or icosidodecahedron. I have not found yet any polytope where the construction loops.