Editing Isogonal figure (section)

==Isogonal polyhedra and 2D tilings==
{| class=wikitable align=right
|+ Isogonal tilings
|[[File:Isogonal snub square tiling1.svg|160px]]
|-
|Distorted [[square tiling]]
|-
|[[File:Distorted truncated square tiling.png|160px]]
|-
|-
|A distorted<BR>[[truncated square tiling]]
|}

An '''isogonal polyhedron''' and 2D tiling has a single kind of vertex. An '''isogonal polyhedron''' with all regular faces is also a '''[[uniform polyhedron]]''' and can be represented by a [[vertex configuration]] notation sequencing the faces around each vertex. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration.

{| class=wikitable width=570
|+ Isogonal polyhedra
!D<sub>3d</sub>, order 12
![[Pyritohedral symmetry|T<sub>h</sub>]], order 24
!colspan=2|[[Octahedral symmetry|O<sub>h</sub>]], order 48
|- valign=top
!4.4.6
!3.4.4.4
!4.6.8
!3.8.8
|- valign=top
|[[File:Cantic snub hexagonal hosohedron2.png|150px]]<BR>A distorted [[hexagonal prism]] (ditrigonal trapezoprism)
|[[File:Cantic snub octahedron.png|140px]]<BR>A distorted [[rhombicuboctahedron]]
|[[File:Truncated rhombicuboctahedron nonuniform.png|140px]]<BR>A shallow [[truncated cuboctahedron]]
|[[File:Cube truncation 1.50.png|140px]]<BR>A hyper-truncated cube
|}

Isogonal polyhedra and 2D tilings may be further classified:

* ''[[Regular polyhedron|Regular]]'' if it is also [[isohedral]] (face-transitive) and [[isotoxal]] (edge-transitive); this implies that every face is the same kind of [[regular polygon]].
* ''[[Quasiregular polyhedron|Quasi-regular]]'' if it is also [[isotoxal]] (edge-transitive) but not [[isohedral]] (face-transitive).
* ''[[Semiregular polyhedron|Semi-regular]]'' if every face is a regular polygon but it is not  [[isohedral]] (face-transitive) or [[isotoxal]] (edge-transitive).  (Definition varies among authors; e.g. some exclude solids with dihedral symmetry, or nonconvex solids.)
* ''[[Uniform polyhedron|Uniform]]'' if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular.
* ''Semi-uniform'' if its elements are also isogonal.
* ''Scaliform'' if all the edges are the same length.
* ''[[Noble polyhedron|Noble]]'' if it is also [[isohedral]] (face-transitive).