Facet (geometry)In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face. To facet a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higher-dimensional polytopes.
Flag (geometry)In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension. More formally, a flag ψ of an n-polytope is a set {F_–1, F_0, ..., F_n} such that F_i ≤ F_i+1 (–1 ≤ i ≤ n – 1) and there is precisely one F_i in ψ for each i, (–1 ≤ i ≤ n). Since, however, the minimal face F_–1 and the maximal face F_n must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces.
Real projective planeIn mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in \mathbb{R}^3 passing through the origin.
