Projective polyhedron

In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of polyhedra, which can be interpreted as spherical polyhedra – tessellations of the sphere, and analogous to toroidal polyhedra – tessellations of the torus.

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".

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.

Examples

The Hemi-cube is a regular projective polyhedron with 3 square faces, 6 edges, and 4 vertices.

The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids:

• Hemi-cube
• Hemi-octahedron
• Hemi-dodecahedron
• Hemi-icosahedron

These can be obtained by taking the quotient of the associated spherical polygon 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.

The tetrahemihexahedron is a projective polyhedron, and the only uniform projective polyhedra that immerses in Euclidean 3-space.

Note that the prefix "hemi-" is also used to refer to hemipolyhedron, which are uniform polyhedra that pass through the center. As these do not define spherical polyhedra (because they pass through the center, which does not map to a defined point on the sphere), they do not define projective polyhedra by the quotient map from 3-space (minus the origin) to the projective plane.

Of these hemipolyhedra, only the tetrahemihexahedron is topologically a projective polyhedron, as can be verified by its Euler characteristic and visually obvious connection to the Roman surface. It is 2-covered by the cuboctahedron, and can be realized as the quotient of the spherical cuboctahedron by the antipodal map. It is the only uniform (traditional) polyhedron that is projective – that is, the only uniform projective polyhedron that immerses in Euclidean three-space as a uniform traditional polyhedron.

Relation with spherical polyhedra

There is a 2-to-1 covering map $S^{2}\to \mathbf {RP} ^{2}$ of the sphere to the projective plane.

This does not precisely yield a Galois connection between spherical polyhedra and projective polyhedra as follows – every projective polyhedron yields a spherical polyhedron (by 2-fold cover), but a spherical polyhedron which is not symmetric through the origin does not define a projective polyhedron.

Spherical polyhedra are related to projective polyhedra (tessallations of the real projective plane) – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin.

For example, the 2-fold cover of the (projective) hemi-cube is the (spherical) cube. hemi-cube: 4 vertices, 3 faces, 6 edges; cube: 8 vertices, 6 faces, 12 edges

They have the same abstract vertex figure.

tetrahedron: 4 vertices, 6 edges, 4 faces; adjunction yields 8 vertices (of the cube)

For tetrahedron it’s trickier – natural adjunction is improper (Stellated octahedron), while triakis octahedron is less natural but proper.

Generalizations

In the context of abstract polytopes, one instead refers to "locally projective polytopes" – see Abstract polytope: Local topology. For example, the 11-cell is a "locally projective polytope", but is not a globally projective polyhedron, nor indeed tessellates any manifold, as it not locally Euclidean, but rather locally projective, as the name indicates.

Projective polytopes can be defined in higher dimension as tessellations of projective space in higher dimensions. Defining k-dimensional projective polytopes in n-dimensional projective space is somewhat trickier, because the usual definition of polytopes in Euclidean space requires taking convex combinations of points, which is not a projective concept, and is infrequently addressed in the literature, but has been defined, such as in (Vives & Mayo 1991).

Symmetry group

The symmetry group of a projective polyhedron is the rotational symmetry group of the covering spherical polyhedron; the full symmetry group of the spherical polyhedron is then just the direct product with reflection through the origin, which is the kernel on passage to projective space.

See also projective orthogonal group and binary polyhedral group.

See also

• Spherical polyhedron
• Toroidal polyhedron

References

• Magnus, Wilhelm (1974), Noneuclidean tesselations and their groups, Academic Press, ISBN 978-0-12465450-1
• Hilbert, David; Cohn-Vossen, S. (1999), Geometry and the imagination, AMS Bookstore, p. 147, ISBN 978-0-82181998-2
• Vives, Gilberto Calvillo; Mayo, Guillermo Lopez (1991). Advances in numerical partial differential equations and optimization. Fifth United States-Mexico Workshop. SIAM. pp. 43–49. ISBN 978-0-89871269-8. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help); Unknown parameter |editors= ignored (|editor= suggested) (help)
• McMullen, Peter; Schulte, Egon (2002), "6C. Projective Regular Polytopes", Abstract Regular Polytopes (1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0 {{citation}}: Unknown parameter |month= ignored (help)
• Schulte, Egon; Weiss, Asia Ivic (2006), "5 Topological classification", Problems on Polytopes, Their Groups, and Realizations, pp. 9–13, arXiv:math/0608397v1