Orientability

Orientability is a property of connected topological manifolds that also applies to all polytopes. Informally, if the polytope is of rank $n$, it is orientable if an $(n - 1)$-dimensional chiral figure moving across the surface of the polytope cannot be transformed into its mirror image. Examples include all convex polytopes, small stellated dodecahedron and small cubicuboctahedron. A non-orientable polytope is simply one that is not orientable. Examples include the tetrahemihexahedron and small rhombihexahedron.

Orientability is an abstract property; it can be determined from the underlying abstract polytope alone. In particular, an abstract polytope is orientable iff its flag graph is. Orientability can also be determined for maps or maniplexes which are not necessarily abstract polytopes.

Non-orientable polytopes have no notion of volume or density.

Neither exotic polytopoids nor polytope compounds are valid connected manifolds, so the notion of orientability does not apply to them. Compounds of orientable polytopes may be called orientable because their surfaces have no orientation reversing paths, and compounds of non-orientable polytopes may be called non-orientable since every point on their surface has a orientation reversing path. However compounds of which mix orientable and non-orientable polytopes have no defined orientability.