by **Klitzing** » Sun Dec 24, 2017 5:17 pm

Just consider the facial planes of the polytope. Provided the polytope has a well-defined center, then these planes usually are off that point. Thus you can clearly say, whether a side is out or in, depending on the respective orientation wrt. that special point. But when a face would happen to run through that point, then this type of orientation breaks down.

The simplest example is the tetrahemihexahedron built from a tetrahedral subset of the faces of an octahedron (which clearly show up an individual orientability) plus the 3 diametral squares (which don't).

A more elaborate definition here consideres the connectivity of the facial planes as well. Think about a small ant crawling apon those facial planes, but which would be unaware of any facial intersection. Then this ant might crawl on the outside of one triangle, then will turn around an edge, thus crawling on the top of a square, tunneling through the square intersections, thereby still remaining on the same side of it. When it comes to the next edge it would have to move on at the inside of the next triangle. Thus when moving on and on, this very ant would travel on either side of every facial plane. That is, the polytope is not orientable, just like the Moebius strip.

Even so the consideration of connectivities in the second case is a bit different from the former case of individual orientabilities, the very argument for an after all cross-over comes down again to those facial planes, which happen to run through the center point.

--- rk