Editing Abstract polytope (section)

==Examples of higher rank==

The set of ''j''-faces (−1 ≤ ''j'' ≤ ''n'') of a traditional ''n''-polytope form an abstract ''n''-polytope.

The concept of an abstract polytope is more general and also includes:
* [[Apeirotope]]s or infinite polytopes, which include [[tessellation]]s (tilings)
* Proper decompositions of unbounded manifolds such as the [[torus]] or [[real projective plane]].
* Many other objects, such as the [[11-cell]] and the [[57-cell]], that cannot be faithfully realized in Euclidean spaces.

===Hosohedra and hosotopes===
[[File:Hexagonal Hosohedron.svg|thumb|A hexagonal [[hosohedron]], realized as a [[spherical polyhedron]].]]
The digon is generalized by the [[hosohedron]] and higher dimensional hosotopes, which can all be realized as [[spherical polyhedra]] – they tessellate the sphere.

===Projective polytopes===
[[Image:Hemicube.svg|thumb|220px|The [[Hemi-cube (geometry)|Hemicube]] may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.]]
Four examples of non-traditional abstract polyhedra are the [[Hemi-cube (geometry)|Hemicube]] (shown), [[Hemi-octahedron]], [[Hemi-dodecahedron]], and the [[Hemi-icosahedron]]. These are the projective counterparts of the [[Platonic solid]]s, and can be realized as (globally) [[projective polyhedra]] – they tessellate the [[real projective plane]].

The hemicube is another example of where vertex notation cannot be used to define a polytope - all the 2-faces and the 3-face have the same vertex set.