Editing Regular polyhedron (section)

==== Regular tilings of the real projective plane ====
Another group of regular polyhedra comprise tilings of the [[real projective plane]]. These include the [[Hemi-cube (geometry)|hemi-cube]], [[hemi-octahedron]], [[hemi-dodecahedron]], and [[hemi-icosahedron]]. They are (globally) [[projective polyhedra]], and are the projective counterparts of the [[Platonic solid]]s. The tetrahedron does not have a projective counterpart as it does not have pairs of parallel faces which can be identified, as the other four Platonic solids do.

{| class=wikitable
|- align=center
|[[File:Hemicube.svg|150px]]<br>[[Hemicube (geometry)|Hemi-cube]]<br>{4,3}
|[[File:Hemioctahedron.png|150px]]<br>[[Hemi-octahedron]]<br>{3,4}
|[[File:Hemi-Dodecahedron2.PNG|150px]]<br>[[Hemi-dodecahedron]]<br>{3,5}
|[[File:Hemi-icosahedron.png|150px]]<br>[[Hemi-icosahedron]]<br>{5,3}
|}

These occur as dual pairs in the same way as the original Platonic solids do. Their Euler characteristics are all 1.