Editing Regular polyhedron (section)

=== 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>
|}