Planar polytope

A polytope of rank n is planar if its vertices lie in n-dimensional Euclidean space and the vertices of each proper element of rank r lie in an r-dimensional affine subspace. That is, each face lies in a plane, each cell lies in a 3-space, etc.

Since antiquity, planarity has been a common condition imposed on the geometrical study of polytopes (as opposed to purely combinatorial studies), including convex, regular, uniform, and CRF polytopes, as well as many Euclidean tilings.

Planarity assists with ensuring that polytopes have defined interiors (although it does not guarantee it, see filling method), and presents one way for polytope definitions to "recurse" so that proper elements are themselves valid polytopes.

One definition of a skew polytope is simply any polytope realized in Euclidean space which is not planar.

Planar realizations

A planar realization is simply a faithful realization of a polytope in Euclidean space that is planar.

It is unclear from the literature how difficult it is to determine whether a general, given abstract polytope has a planar realization, possibly self-intersecting and not necessarily respecting any isomorphisms of the underlying abstract polytope. A special case where the decision problem is straightforward comes from Steinitz's theorem, which states that the 1-skeletons of convex polyhedra are precisely the planar 3-connected graphs.