as facets of noncompact hyperbolic tessellations, as in the heptagonal tiling honeycomb {7,3,3}; they are inscribed in an equidistant surface (a 2-hypercycle), which has two ideal points. Regular tilings of the real projective plane Another group of regular polyhedra comprise tilings of the real projective plane. These include the hemi-cube, hemi-octahedron, hemi-dodecahedron, and hemi-icosahedron. They are (globally) projective polyhedra, and are the projective counterparts of the Platonic solids. The tetrahedron does not have a projective counterpart as it does not have pairs of parallel faces which can be identified, as the other four Platonic solids do. These occur as dual pairs in the same way as the original Platonic solids do. Their Euler characteristics are all 1. Abstract regular polyhedra 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. Five such regular abstract polyhedra, which can not be realised faithfully, were identified by H. S. M. Coxeter in his book Regular Polytopes (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987). All five have C2×S5 symmetry but can only be realised with half the symmetry, that is C2×A5 or icosahedral symmetry. They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those