A regular 5-polytope formula_15 exists only if formula_2 and formula_21 are regular 4-polytopes. The space it fits in is based on the expression:  Enumeration of these constraints produce "3" convex polytopes, "zero" nonconvex polytopes, "3" 4-space tessellations, and "5" hyperbolic 4-space tessellations. There are no non-convex regular polytopes in five dimensions or higher. In dimensions 5 and higher, there are only three kinds of convex regular polytopes. There are also improper cases where some numbers in the Schläfli symbol are 2. For example, {p,q,r...2} is an improper regular spherical polytope whenever {p,q,r...} is a regular spherical polytope, and {2...p,q,r} is an improper regular spherical polytope whenever {...p,q,r} is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as {p,q...2...y,z}. There are no non-convex regular polytopes in five dimensions or higher. A projective regular ("n"+1)-polytope exists when an original regular "n"-spherical tessellation, {p,q...}, is centrally symmetric. Such a polytope is named hemi-{p,q...}, and contain half as many elements. Coxeter gives a symbol {p,q...}/2, while McMullen writes {p,q...}h/2 with "h" as the coxeter number. Even-sided regular polygons have hemi-"2n"-gon projective polygons, {2p}/2. There are 4 regular projective polyhedra related to 4 of 5 Platonic solids. The hemi-cube and hemi-octahedron generalize as hemi-"n"-cubes and hemi-"n"-orthoplexes in any dimensions. In 4-dimensions 5 of 6 convex regular 4-polytopes generate projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell. There are only 2 convex regular projective hemi-polytopes in dimensions 5 or higher.