In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane.[1] These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.
Projective polyhedra are also referred to as elliptic tessellations[2] or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling,[3] a synonym for "spherical polyhedron". However, the term elliptic geometry applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.
As cellular decompositions of the projective plane, they have Euler characteristic 1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with locally projective polyhedra, which are defined in the theory of abstract polyhedra.
Non-overlapping projective polyhedra (density 1) correspond to spherical polyhedra (equivalently, convex polyhedra) with central symmetry. This is elaborated and extended below in relation with spherical polyhedra and relation with traditional polyhedra.
^Schulte, Egon; Weiss, Asia Ivic (2006), "5 Topological classification", Problems on Polytopes, Their Groups, and Realizations, pp. 9–13, arXiv:math/0608397v1, Bibcode:2006math......8397S
^Coxeter, Harold Scott Macdonald (1970). Twisted honeycombs. CBMS regional conference series in mathematics (4). AMS Bookstore. p. 11. ISBN 978-0-8218-1653-0.
^Magnus, Wilhelm (1974), Noneuclidean tesselations and their groups, Academic Press, p. 65, ISBN 978-0-12-465450-1
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