Editing Regular polyhedron (section)

==== 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>
|}