Editing Platonic solid (section)

== Related polyhedra and polytopes ==
=== Uniform polyhedra ===
There exist four regular polyhedra that are not convex, called [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]]. These all have [[icosahedral symmetry]] and may be obtained as [[stellation]]s of the dodecahedron and the icosahedron.

{| class="wikitable floatright" style="text-align:center"
|- style="vertical-align:bottom;"
| [[Image:Cuboctahedron.svg|120px]]<br />[[cuboctahedron]]
| [[Image:Icosidodecahedron.svg|120px]]<br />[[icosidodecahedron]]
|}

The next most regular convex polyhedra after the Platonic solids are the [[cuboctahedron]], which is a [[rectification (geometry)|rectification]] of the cube and the octahedron, and the [[icosidodecahedron]], which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both ''quasi-regular'', meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen [[Archimedean solid]]s, which are the convex [[uniform polyhedron|uniform polyhedra]] with polyhedral symmetry. Their duals, the [[rhombic dodecahedron]] and [[rhombic triacontahedron]], are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen [[Catalan solid]]s.

The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of [[regular polygon|regular]] or [[star polygon]]s for faces. These include all the polyhedra mentioned above together with an infinite set of [[prism (geometry)|prisms]], an infinite set of [[antiprism]]s, and 53 other non-convex forms.

The [[Johnson solid]]s are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex [[deltahedron|deltahedra]], which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)

=== 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}]]
|}

=== Higher dimensions ===
{{Further|List of regular polytopes}}
{| class="wikitable floatright" style="text-align:center; max-width: 22em"
|-
! {{nowrap|Number of}} dimensions
! {{nowrap|Number of convex}} regular polytopes
|-
| 0 || 1
|-
| 1 || 1
|-
| 2 || &infin;
|-
| '''3''' || '''5'''
|-
| 4 || 6
|-
| &gt; 4 || 3
|}

In more than three dimensions, polyhedra generalize to [[polytope]]s, with higher-dimensional convex [[regular polytope]]s being the equivalents of the three-dimensional Platonic solids.

In the mid-19th century the Swiss mathematician [[Ludwig Schläfli]] discovered the four-dimensional analogues of the Platonic solids, called [[convex regular 4-polytope]]s. There are exactly six of these figures; five are analogous to the Platonic solids : [[5-cell]] as {3,3,3}, [[16-cell]] as {3,3,4}, [[600-cell]] as {3,3,5}, [[tesseract]] as {4,3,3}, and [[120-cell]] as {5,3,3}, and a sixth one, the self-dual [[24-cell]], {3,4,3}.

In all dimensions higher than four, there are only three convex regular polytopes: the [[simplex]] as {3,3,...,3}, the [[hypercube]] as {4,3,...,3}, and the [[cross-polytope]] as {3,3,...,4}.{{sfn|Coxeter|1973|p=136}} In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.