Editing Regular polyhedron (section)

=== Regular polyhedra in non-Euclidean and other spaces ===
Studies of [[non-Euclidean]] ([[hyperbolic space|hyperbolic]] and [[elliptic space|elliptic]]) and other spaces such as [[complex affine space|complex spaces]], discovered over the preceding century, led to the discovery of more new polyhedra such as [[complex polytope|complex polyhedra]] which could only take regular geometric form in those spaces.

==== Regular polyhedra in hyperbolic space ====

[[File:633 honeycomb one cell horosphere.png|thumb|The [[hexagonal tiling honeycomb]], {6,3,3}, has [[hexagonal tiling]], {6,3}, facets with vertices on a [[horosphere]]. One such facet is shown in as seen in this [[Poincaré disk model]].]]
In H<sup>3</sup> [[hyperbolic space]], [[paracompact regular honeycomb]]s have Euclidean tiling [[Facet (geometry)|facets]] and [[vertex figure]]s that act like finite polyhedra. Such tilings have an [[angle defect]] that can be closed by bending one way or the other. If the tiling is properly scaled, it will ''close'' as an [[Asymptote|asymptotic limit]] at a single [[ideal point]]. These Euclidean tilings are inscribed in a [[horosphere]] just as polyhedra are inscribed in a sphere (which contains zero ideal points). The sequence extends when hyperbolic tilings are themselves used as facets of noncompact hyperbolic tessellations, as in the [[heptagonal tiling honeycomb]] {7,3,3}; they are inscribed in an equidistant surface (a 2-[[hypercycle (hyperbolic geometry)|hypercycle]]), which has two ideal points.

==== 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.