Editing Isogonal figure

{{Short description|Polytope or tiling whose vertices are identical}}
{{For|[[graph theory]]|vertex-transitive graph}}

In [[geometry]], a [[polytope]] (e.g. a [[polygon]] or [[polyhedron]]) or a [[Tessellation|tiling]]  is '''isogonal''' or '''vertex-transitive''' if all its [[vertex (geometry)|vertices]] are equivalent under the [[Symmetry|symmetries]] of the figure. This implies that each vertex is surrounded by the same kinds of [[face (geometry)|face]] in the same or reverse order, and with the same [[Dihedral angle|angles]] between corresponding faces.

Technically, one says that for any two vertices there exists a symmetry of the polytope [[Map (mathematics)|mapping]] the first [[isometry|isometrically]] onto the second. Other ways of saying this are that the [[automorphism group|group of automorphisms]] of the polytope ''[[Group action#Remarkable properties of actions|acts transitively]]'' on its vertices, or that the vertices lie within a single ''[[symmetry orbit]]''.

All vertices of a finite {{mvar|n}}-dimensional isogonal figure exist on an [[n-sphere|{{math|(''n''−1)}}-sphere]].<ref>{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.1007/PL00009307
 | issue = 1
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1453440
 | pages = 13–52
 | title = Isogonal prismatoids
 | volume = 18
 | year = 1997}}</ref>

The term '''isogonal''' has long been used for polyhedra. '''Vertex-transitive''' is a synonym borrowed from modern ideas such as [[symmetry group]]s and [[graph theory]].

The [[Elongated square gyrobicupola|pseudorhombicuboctahedron]]{{snd}}which is ''not'' isogonal{{snd}}demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.

==Isogonal polygons and apeirogons==
{| class=wikitable align=right
|[[File:Uniform apeirogon.png|200px]]
|-
|[[File:isogonal_apeirogon_linear.png|200px]]
|-
!Isogonal [[apeirogon]]s
|-
|[[File:isogonal_apeirogon.png|200px]]
|-
|[[File:Isogonal_apeirogon2.png|200px]]
|-
|[[File:Isogonal_apeirogon2a.png|200px]]
|-
|[[File:Isogonal_apeirogon2b.png|200px]]
|-
|[[File:Isogonal_apeirogon2c.png|200px]]
|-
|[[File:Isogonal_apeirogon2d.png|200px]]
|-
!Isogonal [[skew apeirogon]]s
|}
All [[regular polygons]], [[apeirogon]]s and [[regular star polygon]]s are ''isogonal''. The [[Dual polygon|dual]] of an isogonal polygon is an [[isotoxal polygon]].

Some even-sided polygons and [[apeirogon]]s which alternate two edge lengths, for example a [[rectangle]], are ''isogonal''.

All planar isogonal 2''n''-gons have [[dihedral symmetry]] (D<sub>''n''</sub>, ''n''&thinsp;=&thinsp;2,&thinsp;3,&thinsp;...) with reflection lines across the mid-edge points.
{| class=wikitable width=560
!D<sub>2</sub>
!D<sub>3</sub>
!D<sub>4</sub>
!D<sub>7</sub>
|-
|[[File:Crossed rectangles.png|200px]]<BR>Isogonal [[rectangle]]s and [[crossed rectangle]]s sharing the same [[vertex arrangement]]
|[[File:Regular truncation 3 0.75.svg|120px]]<BR>Isogonal [[hexagram#Other hexagrams|hexagram]] with 6 identical vertices and 2 edge lengths.<ref>Coxeter, The Densities of the Regular Polytopes II, p54-55, "hexagram" vertex figure of h{5/2,5}.</ref>
|[[File:Vertex-transitive-octagon.svg|140px]]<BR>Isogonal convex [[octagon]] with blue and red radial lines of reflection
|[[File:Regular polygon truncation 7 3.svg|150px]]<BR>Isogonal "star" [[tetradecagon]] with one vertex type, and two edge types<ref>''The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History'', (1994), ''Metamorphoses of polygons'', [[Branko Grünbaum]], Figure 1. Parameter ''t''=2.0</ref>
|}
{{-}}

==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).

==''N'' dimensions: Isogonal polytopes and tessellations==
These definitions can be extended to higher-dimensional [[polytope]]s and [[Honeycomb (geometry)|tessellations]].  All [[uniform polytope]]s are ''isogonal'', for example, the [[uniform 4-polytope]]s and [[convex uniform honeycomb]]s.

The [[Dual polytope|dual]] of an isogonal polytope is an [[isohedral figure]], which is transitive on its [[Facet (geometry)|facets]].

==''k''-isogonal and ''k''-uniform figures==
A polytope or tiling may be called '''''k''-isogonal''' if its vertices form ''k'' transitivity classes. A more restrictive term, '''''k''-uniform''' is defined as a ''k-isogonal figure'' constructed only from [[regular polygon]]s. They can be represented visually with colors by different [[uniform coloring]]s.

{| class=wikitable width=600
|- valign=top
|[[File:Truncated rhombic dodecahedron2.png|200px]]<BR>This [[truncated rhombic dodecahedron]] is '''2-isogonal''' because it contains two transitivity classes of vertices. This polyhedron is made of [[Square (geometry)|squares]] and flattened [[hexagon]]s.
|[[File:2-uniform 11.png|200px]]<BR>This [[Euclidean tilings of regular polygons#2-uniform tilings|demiregular tiling]] is also '''2-isogonal''' (and '''2-uniform'''). This tiling is made of [[equilateral triangle]] and regular [[hexagon]]al faces.
|[[File:Enneagram 9-4 icosahedral.svg|200px]]<BR>2-isogonal 9/4 [[Enneagram (geometry)|enneagram]] (face of the [[final stellation of the icosahedron]])
|}

==See also==
* [[Edge-transitive]] (Isotoxal figure)
* [[Face-transitive]] (Isohedral figure)

==References==
{{Reflist}}
* Peter R. Cromwell, ''Polyhedra'', Cambridge University Press 1997, {{ISBN|0-521-55432-2}}, p.&nbsp;369 Transitivity
* {{Cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | author2-link=G.C. Shephard | title=Tilings and Patterns | publisher=W. H. Freeman and Company | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (p.&nbsp;33 ''k-isogonal'' tiling, p.&nbsp;65 ''k-uniform tilings'')

==External links==
*{{MathWorld | urlname=Vertex-TransitiveGraph | title=Vertex-transitive graph }}
*[http://bulatov.org/polyhedra/mosaic2000/ Isogonal Kaleidoscopical Polyhedra] [[Vladimir L. Bulatov]], Physics Department, Oregon State University, Corvallis, Presented at Mosaic2000, Millennial Open Symposium on the Arts and Interdisciplinary Computing, 21–24 August 2000, Seattle, WA [http://www.bulatov.org/polyhedra//mosaic2000/kaleido_poly/kaleido_frames.html VRML models]
* [https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm Steven Dutch uses the term k-uniform for enumerating k-isogonal tilings]
* [http://probabilitysports.com/tilings.html List of n-uniform tilings]
*{{MathWorld | urlname=DemiregularTessellation| title=Demiregular tessellations}} (Also uses term k-uniform for k-isogonal)

{{polygons}}

{{DEFAULTSORT:Isogonal Figure}}
[[Category:Polyhedra]]
[[Category:Polytopes]]