Editing Regular polyhedron (section)

=== Spherical polyhedra ===
{{main|Spherical polyhedron}}

The usual five regular polyhedra can also be represented as spherical tilings (tilings of the [[sphere]]):

{| class=wikitable width=640
|- align=center
|[[File:Uniform tiling 332-t2.svg|100px]]<br>[[Tetrahedron]]<br>{3,3}
|[[File:Uniform tiling 432-t0.png|100px]]<br>[[Cube]]<br>{4,3}
|[[File:Uniform tiling 432-t2.png|100px]]<br>[[Octahedron]]<br>{3,4}
|[[File:Uniform tiling 532-t0.png|100px]]<br>[[Dodecahedron]]<br>{5,3}
|[[File:Uniform tiling 532-t2.png|100px]]<br>[[regular icosahedron|Icosahedron]]<br>{3,5}
|}
{| class=wikitable width=640
|- align=center
|[[File:Small stellated dodecahedron tiling.png|100px]]<br>[[Small stellated dodecahedron]]<br>{5/2,5}
|[[File:Great dodecahedron tiling.svg|100px]]<br>[[Great dodecahedron]]<br>{5,5/2}
|[[File:Great stellated dodecahedron tiling.svg|100px]]<br>[[Great stellated dodecahedron]]<br>{5/2,3}
|[[File:Great icosahedron tiling.svg|100px]]<br>[[Great icosahedron]]<br>{3,5/2}
|}

==== Regular polyhedra that can only exist as spherical polyhedra ====
{{see also|Hosohedron|Dihedron}}
For a regular polyhedron whose Schläfli symbol is {''m'',&nbsp;''n''}, the number of polygonal faces may be found by:

:<math>N_2=\frac{4n}{2m+2n-mn}</math>

The [[Platonic solid]]s known to antiquity are the only integer solutions for ''m'' ≥ 3 and ''n'' ≥ 3. The restriction ''m'' ≥ 3 enforces that the polygonal faces must have at least three sides.

When considering polyhedra as a [[spherical tiling]], this restriction may be relaxed, since [[digon]]s (2-gons) can be represented as spherical lunes, having non-zero [[area (geometry)|area]]. Allowing ''m'' = 2 admits a new infinite class of regular polyhedra, which are the [[hosohedron|hosohedra]]. On a spherical surface, the regular polyhedron {2,&nbsp;''n''} is represented as ''n'' abutting lunes, with interior angles of 2{{pi}}/''n''. All these lunes share two common vertices.<ref name=cox>Coxeter, ''Regular polytopes'', p. 12</ref>

A regular [[dihedron]], {''n'',&nbsp;2}<ref name=cox /> (2-hedron) in three-dimensional [[Euclidean space]] can be considered a [[degeneracy (mathematics)|degenerate]] [[Prism (geometry)|prism]] consisting of two (planar) ''n''-sided [[polygon]]s connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with two [[line segment]]s. However, as a [[spherical tiling]], a dihedron can exist as nondegenerate form, with two ''n''-sided faces covering the sphere, each face being a [[Sphere|hemisphere]], and vertices around a [[great circle]]. It is ''regular'' if the vertices are equally spaced.

{| class=wikitable
|- align=center valign=bottom
|[[File:Digonal dihedron.png|100px]]<br>[[Digon]]al dihedron<br>{2,2}
|[[File:Trigonal dihedron.png|100px]]<br>[[Triangle|Trigonal]] dihedron<br>{3,2}
|[[File:Tetragonal dihedron.png|100px]]<br>[[Square]] dihedron<br>{4,2}
|[[File:Pentagonal dihedron.png|100px]]<br>[[Pentagon]]al dihedron<br>{5,2}
|[[File:Hexagonal dihedron.png|100px]]<br>[[Hexagon]]al dihedron<br>{6,2}
|...
|{''n'',2}
|- align=center valign=bottom
|[[File:Digonal dihedron.png|100px]]<br>Digonal hosohedron<br>{2,2}
|[[File:Trigonal hosohedron.png|100px]]<br>Trigonal hosohedron<br>{2,3}
|[[File:Spherical square hosohedron.svg|100px]]<br>Square hosohedron<br>{2,4}
|[[File:Spherical pentagonal hosohedron.svg|100px]]<br>Pentagonal hosohedron<br>{2,5}
|[[File:Hexagonal hosohedron.png|100px]]<br>Hexagonal hosohedron<br>{2,6}
|...
|{2,''n''}
|}

The hosohedron {2,''n''} is dual to the dihedron {''n'',2}. Note that when ''n'' = 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron. All of these have Euler characteristic 2.