Editing Regular polyhedron (section)

== Further generalisations ==
The 20th century saw a succession of generalisations of the idea of a regular polyhedron, leading to several new classes.

=== Regular skew apeirohedra ===

{{Main|Regular skew apeirohedron}}

In the first decades, Coxeter and Petrie allowed "saddle" vertices with alternating ridges and valleys, enabling them to construct three infinite folded surfaces which they called [[regular skew polyhedron|regular skew polyhedra]].<ref>[[Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, {{isbn|0-486-40919-8}} (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)</ref> Coxeter offered a modified [[Schläfli symbol]] {l,m|n} for these figures, with {l,m} implying the [[vertex figure]], with ''m'' regular ''l''-gons around a vertex. The ''n'' defines ''n''-gonal ''holes''. Their vertex figures are [[regular skew polygon]]s, vertices zig-zagging between two planes.

{| class="wikitable"
!colspan=3| Infinite regular skew polyhedra in 3-space (partially drawn)
|- align=center
||[[Image:mucube external.png|150px]]<br>{4,6|4}
|[[Image:muoctahedron external.png|150px]]<br>{6,4|4}
|[[Image:mutetrahedron external.png|150px]]<br>{6,6|3}
|}

=== Regular skew polyhedra ===
{{Main|Regular skew polyhedron}}

Finite regular skew polyhedra exist in 4-space. These finite regular skew polyhedra in 4-space can be seen as a subset of the faces of [[uniform 4-polytope]]s. They have planar [[regular polygon]] faces, but [[regular skew polygon]] [[vertex figure]]s.

Two dual solutions are related to the [[5-cell]], two dual solutions are related to the [[24-cell]], and an infinite set of self-dual [[duoprism]]s generate regular skew polyhedra as {4, 4 {{pipe}} n}. In the infinite limit these approach a [[duocylinder]] and look like a [[torus]] in their [[stereographic projection]]s into 3-space.

{| class=wikitable
|+ Finite regular skew polyhedra in 4-space
|-
!colspan=4|Orthogonal [[Coxeter plane]] projections
!rowspan=2|[[Stereographic projection]]
|-
!colspan=2| A<sub>4</sub>
!colspan=2| F<sub>4</sub>
|-
|[[File:4-simplex t03.svg|150px]]
|[[File:4-simplex t12.svg|150px]]
|[[File:24-cell t03 F4.svg|150px]]
|[[File:24-cell t12 F4.svg|150px]]
|[[File:Clifford-torus.gif|150px]]
|-
![[Runcinated 5-cell#Related skew polyhedron|{4, 6 {{pipe}} 3}]]
![[Truncated 5-cell#Related skew polyhedron|{6, 4 {{pipe}} 3}]]
![[Runcinated 24-cell#Related regular skew polyhedron|{4, 8 {{pipe}} 3}]]
![[Truncated 24-cells#Related regular skew polyhedron|{8, 4 {{pipe}} 3}]]
![[Duoprism#Related polytopes|{4, 4 {{pipe}} n}]]
|-
!30 [[square|{4}]] faces<BR>60 edges<BR>20 vertices
!20 [[hexagon|{6}]] faces<BR>60 edges<BR>30 vertices
!288 {4} faces<BR>576 edges<BR>144 vertices
!144 [[octagon|{8}]] faces<BR>576 edges<BR>288 vertices
!{{math|''n''<sup>2</sup>}} {4} faces<BR>{{math|2''n''<sup>2</sup>}} edges<BR>{{math|''n''<sup>2</sup>}} vertices
|}

=== 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.

=== Abstract regular polyhedra ===
{{See|Abstract regular polytope}}

By now, polyhedra were firmly understood as three-dimensional examples of more general ''[[polytope]]s'' in any number of dimensions. The second half of the century saw the development of abstract algebraic ideas such as [[Polyhedral combinatorics]], culminating in the idea of an [[abstract polytope]] as a [[partially ordered set]] (poset) of elements. The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the ''null polytope'' or empty set. These abstract elements can be mapped into ordinary space or ''realised'' as geometrical figures. Some abstract polyhedra have well-formed or ''faithful'' realisations, others do not. A ''flag'' is a connected set of elements of each dimension – for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be ''regular'' if its combinatorial symmetries are transitive on its flags – that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.

Five such regular abstract polyhedra, which can not be realised faithfully, were identified by [[H. S. M. Coxeter]] in his book ''[[Regular Polytopes]]'' (1977) and again by [[J. M. Wills]] in his paper "The combinatorially regular polyhedra of index 2" (1987). All five have C<sub>2</sub>×S<sub>5</sub> symmetry but can only be realised with half the symmetry, that is C<sub>2</sub>×A<sub>5</sub> or icosahedral symmetry.<ref>[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)], David A. Richter</ref><ref>{{cite arXiv | eprint=1005.4911 | last1=Cutler | first1=Anthony M. | last2=Schulte | first2=Egon | title=Regular Polyhedra of Index Two, I | date=2010 | class=math.MG }}</ref><ref>[https://www.researchgate.net/publication/225386108_Regular_Polyhedra_of_Index_Two_II Regular Polyhedra of Index Two, II]  Beitrage zur Algebra und Geometrie 52(2):357–387 · November 2010, Table 3, p.27</ref> They are all topologically equivalent to [[toroid]]s. Their construction, by arranging ''n'' faces around each vertex, can be repeated indefinitely as tilings of the [[Hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]]. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.

:{| class="wikitable" width=400
|- align=center
! Polyhedron
|[[Image:DU36 medial rhombic triacontahedron.png|100px]]<br>[[Medial rhombic triacontahedron]]
|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|[[Image:DU41 medial triambic icosahedron.png|100px]]<br>[[Medial triambic icosahedron]]
|[[Image:Ditrigonal dodecadodecahedron.png|100px]]<br>[[Ditrigonal dodecadodecahedron]]
|[[Image:Excavated dodecahedron.png|100px]]<br>[[Excavated dodecahedron]]
|- align=center
!Type
||Dual {5,4}<sub>6</sub> ||{5,4}<sub>6</sub> ||Dual of {5,6}<sub>4</sub> ||{5,6}<sub>4</sub> || {6,6}<sub>6</sub>
|- align=center
!(''v'',''e'',''f'')
|(24,60,30) ||(30,60,24) ||(24,60,20) ||(20,60,24) ||(20,60,20)
|- align=center
![[Vertex figure]]
|{5}, {5/2}<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|(5.5/2)<sup>2</sup><br>[[File:Dodecadodecahedron vertfig.png|60px]]
|{5}, {5/2}<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|(5.5/3)<sup>3</sup><br>[[File:Ditrigonal dodecadodecahedron vertfig.png|60px]]
|[[File:Medial triambic icosahedron face.svg|60px]]
|- align=center valign=top
!Faces
|30 rhombi<br>[[File:Rhombus definition2.svg|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<br>[[File:Medial triambic icosahedron face.svg|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular pentagon.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagrams<br>[[File:Star hexagon face.png|60px]]
|- align=center
! Tiling
|[[Image:Uniform tiling 45-t0.png|100px]]<br>[[Order-5 square tiling|{4, 5}]]
|[[Image:Uniform tiling 552-t1.png|100px]]<br>[[Order-4 pentagonal tiling|{5, 4}]]
|[[Image:Uniform tiling 65-t0.png|100px]]<br>[[Order-5 hexagonal tiling|{6, 5}]]
|[[Image:Uniform tiling 553-t1.png|100px]]<br>[[Order-6 pentagonal tiling|{5, 6}]]
|[[Image:Uniform tiling 66-t2.png|100px]]<br>[[Order-6 hexagonal tiling|{6, 6}]]
|- align=center
! [[Euler characteristic|χ]]
| −6
| −6
| −16
| −16
| −20
|}

==== Petrie dual====
{{main|Petrie dual}}
The [[Petrie dual]] of a regular polyhedron is a [[Regular map (graph theory)|regular map]] whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set of [[skew polygon|skew]] [[Petrie polygon]]s.<ref>{{citation|title=Abstract Regular Polytopes|volume=92|series=Encyclopedia of Mathematics and its Applications|first1=Peter|last1=McMullen|first2=Egon|last2=Schulte|publisher=Cambridge University Press|year=2002|isbn=9780521814966|page=192|url=https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA192}}</ref>

{| class=wikitable
|+ Regular petrials
!Name
!Petrial tetrahedron<BR>
!Petrial cube
!Petrial octahedron
!Petrial dodecahedron
!Petrial icosahedron
|- align=center
!Symbol
|{3,3}<sup>{{pi}}</sup>
|{4,3}<sup>{{pi}}</sup>
|{3,4}<sup>{{pi}}</sup>
|{5,3}<sup>{{pi}}</sup>
|{3,5}<sup>{{pi}}</sup>
|- align=center
!(''v'',''e'',''f''), [[Euler characteristic|''&chi;'']]
|(4,6,3), ''&chi;'' = 1||(8,12,4), ''&chi;'' = 0||(6,12,4), ''&chi;'' = −2||(20,30,6), ''&chi;'' = −4||(12,30,6), ''&chi;'' = −12
|- align=center
!rowspan=2|Faces
|rowspan=2|3 skew squares<br/>[[File:Face_of_petrial_tetrahedron.gif|120px]]
|colspan=2 style="border-bottom-style:none;"|4 skew hexagons
|colspan=2 style="border-bottom-style:none;"|6 skew decagons
|- align=center
|style="border-top-style:none;"|[[File:Face_of_petrial_cube.gif|120px]]
|style="border-top-style:none;"|[[File:Face_of_petrial_octahedron.gif|120px]]
|style="border-top-style:none;"|[[File:Face_of_petrial_dodecahedron.gif|120px]]
|style="border-top-style:none;"|[[File:Face_of_petrial_icosahedron.gif|120px]]
|- align=center
!Image
|[[File:Tetrahedron_3_petrie_polygons.png|120px]]
|[[File:Cube_4_petrie_polygons.png|120px]]
|[[File:Octahedron_4_petrie_polygons.png|120px]]
|[[File:Petrial_dodecahedron.png|120px]]
|[[File:petrial_icosahedron.png|120px]]
|- align=center
!Animation
|[[File:Petrial_tetrahedron.gif|120px]]
|[[File:Petrial_cube.gif|120px]]
|[[File:Petrial octahedron.gif|120px]]
|[[File:Petrial_dodecahedron.gif|120px]]
|[[File:petrial_icosahedron.gif|120px]]
|- align=center valign=bottom
!Related<BR>figures
|[[File:Hemicube.svg|120px]]<BR>{4,3}<sub>3</sub> = [[hemi-cube (geometry)|{4,3}/2]] = {4,3}<sub>(2,0)</sub>
|[[File:Regular map 6-3 2-0.png|120px]]<BR>{6,3}<sub>3</sub> = {6,3}<sub>(2,0)</sub>
|[[File:Regular_map_6_4-3_pattern.png|120px]]<BR>{6,4}<sub>3</sub> = {6,4}<sub>(4,0)</sub>
|{10,3}<sub>5</sub>
|{10,5}<sub>3</sub>
|}

=== 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.