Editing Platonic solid (section)

=== Regular tessellations ===
{| class="wikitable floatright"
|+ Regular spherical tilings
! colspan=5 | Platonic
|-
|[[File:Uniform tiling 332-t0-1-.svg|60px]]
|[[File:Uniform tiling 432-t0.png|60px]]
|[[File:Uniform tiling 432-t2.png|60px]]
|[[File:Uniform tiling 532-t0.png|60px]]
|[[File:Uniform tiling 532-t2.png|60px]]
|-
!{3,3}
!{4,3}
!{3,4}
!{5,3}
!{3,5}
|-
! colspan=5 | Regular dihedral
|-
|[[Image:Digonal dihedron.png|60px]]
|[[Image:Trigonal dihedron.png|60px]]
|[[Image:Tetragonal dihedron.png|60px]]
|[[Image:Pentagonal dihedron.png|60px]]
|[[Image:Hexagonal dihedron.png|60px]]
|-
!{2,2}
!{3,2}
!{4,2}
!{5,2}
!{6,2}...
|-
! colspan=5 | Regular hosohedral
|-
|[[Image:Spherical digonal hosohedron.svg|60px]]
|[[Image:Spherical trigonal hosohedron.svg|60px]]
|[[Image:Spherical square hosohedron.svg|60px]]
|[[Image:Spherical pentagonal hosohedron.svg|60px]]
|[[Image:Spherical hexagonal hosohedron.svg|60px]]
|-
!{2,2}
!{2,3}
!{2,4}
!{2,5}
!{2,6}...
|}

The three [[Euclidean tilings by convex regular polygons#Regular tilings|regular tessellation]]s of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the [[sphere]]. This is done by projecting each solid onto a concentric sphere. The faces project onto regular [[spherical polygon]]s which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the [[hosohedra]], {2,''n''} with 2 vertices at the poles, and [[Lune (mathematics)|lune]] faces, and the dual [[dihedra]], {''n'',2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.

Every regular tessellation of the sphere is characterized by a pair of integers {''p'',&nbsp;''q''} with {{sfrac|1|''p''}}&nbsp;+&nbsp;{{sfrac|1|''q''}}&nbsp;>&nbsp;{{sfrac|1|2}}. Likewise, a regular tessellation of the plane is characterized by the condition {{sfrac|1|''p''}}&nbsp;+&nbsp;{{sfrac|1|''q''}}&nbsp;=&nbsp;{{sfrac|1|2}}. There are three possibilities:

{| class=wikitable
|+ The three regular tilings of the Euclidean plane
|[[File:Uniform tiling 44-t0.svg|100px]]
|[[File:Uniform tiling 63-t2-red.svg|100px]]
|[[File:Uniform tiling 63-t0.svg|100px]]
|-
! [[square tiling|{4, 4}]]
! [[triangular tiling|{3, 6}]]
! [[hexagonal tiling|{6, 3}]]
|}
In a similar manner, one can consider regular tessellations of the [[hyperbolic geometry|hyperbolic plane]]. These are characterized by the condition {{sfrac|1|''p''}}&nbsp;+&nbsp;{{sfrac|1|''q''}}&nbsp;<&nbsp;{{sfrac|1|2}}. There is an infinite family of such tessellations.
{| class=wikitable
|+ Example regular tilings of the hyperbolic plane
|[[File:H2-5-4-dual.svg|100px]]
|[[File:H2-5-4-primal.svg|100px]]
|[[File:Heptagonal tiling.svg|100px]]
|[[File:Order-7 triangular tiling.svg|100px]]
|-
! [[Order-4 pentagonal tiling|{5, 4}]]
! [[Order-5 square tiling|{4, 5}]]
! [[Heptagonal tiling|{7, 3}]]
! [[Order-7 triangular tiling|{3, 7}]]
|}