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Concept
Projective polyhedron
Summary
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. 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 or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, 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.
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra:
Hemi-cube, {4,3}/2
Hemi-octahedron, {3,4}/2
Hemi-dodecahedron, {5,3}/2
Hemi-icosahedron, {3,5}/2
Hemi-dihedron, {2p,2}/2, p>=1
Hemi-hosohedron, {2,2p}/2, p>=1
These can be obtained by taking the quotient of the associated spherical polyhedron by the antipodal map (identifying opposite points on the sphere).
On the other hand, the tetrahedron does not have central symmetry, so there is no "hemi-tetrahedron". See relation with spherical polyhedra below on how the tetrahedron is treated.
Note that the prefix "hemi-" is also used to refer to hemipolyhedra, which are uniform polyhedra having some faces that pass through the center of symmetry.
Official source
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