The Classification of Rank 4 Locally Projective Polytopes and Their Quotients

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular polytopes. All 156 of the latter have hemidodecahedral facets or hemi-icosahedral vertex figures. It is noted that, remarkably, every rank 4 locally projective section regular polytope is finite. The article gives a survey of the literature of locally projective polytopes and their quotients, and fills one small gap in the classification in rank 4.