Editing Regular polyhedron (section)

=== 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
|}