By now, polyhedra were firmly understood as three-dimensional examples of more general "polytopes" in any number of dimensions. The second half of the century saw the development of abstract algebraic ideas such as Polyhedral combinatorics, culminating in the idea of an abstract polytope as a partially ordered set (poset) of elements. The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the "null polytope" or empty set. These abstract elements can be mapped into ordinary space or "realised" as geometrical figures. Some abstract polyhedra have well-formed or "faithful" realisations, others do not. A "flag" is a connected set of elements of each dimension – for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be "regular" if its combinatorial symmetries are transitive on its flags – that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.