Editing Abstract polytope (section)

===The 11-cell and the 57-cell===

The [[11-cell]], discovered independently by [[H. S. M. Coxeter]] and [[Branko Grünbaum]], is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a ''locally'' projective polytope. It is self-dual and universal: it is the ''only'' polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.

The [[57-cell]] is also self-dual, with hemi-dodecahedral facets. It was discovered by H. S. M. Coxeter shortly after the discovery of the 11-cell. Like the 11-cell, it is also universal, being the only polytope with hemi-dodecahedral facets and hemi-icosahedral vertex figures. On the other hand, there are many other polytopes with hemi-dodecahedral facets and Schläfli type {5,3,5}. The universal polytope with hemi-dodecahedral facets and icosahedral (not hemi-icosahedral) vertex figures is finite, but very large, with 10006920 facets and half as many vertices.