Ingeometry, apolytope (e.g. apolygon orpolyhedron) or atiling isisogonal orvertex-transitive if all itsvertices are equivalent under thesymmetries of the figure. This implies that each vertex is surrounded by the same kinds offace in the same or reverse order, and with the sameangles between corresponding faces.
Technically, one says that for any two vertices there exists a symmetry of the polytopemapping the firstisometrically onto the second. Other ways of saying this are that thegroup of automorphisms of the polytopeacts transitively on its vertices, or that the vertices lie within a singlesymmetry orbit.
All vertices of a finiten-dimensional isogonal figure exist on an(n−1)-sphere.[1]
The termisogonal has long been used for polyhedra.Vertex-transitive is a synonym borrowed from modern ideas such assymmetry groups andgraph theory.
Thepseudorhombicuboctahedron – which isnot isogonal – 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.
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| Isogonalapeirogons |
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| Isogonalskew apeirogons |
Allregular polygons,apeirogons andregular star polygons areisogonal. Thedual of an isogonal polygon is anisotoxal polygon.
Some even-sided polygons andapeirogons which alternate two edge lengths, for example arectangle, areisogonal.
All planar isogonal 2n-gons havedihedral symmetry (Dn,n = 2, 3, ...) with reflection lines across the mid-edge points.
| D2 | D3 | D4 | D7 |
|---|---|---|---|
 Isogonalrectangles andcrossed rectangles sharing the samevertex arrangement |  Isogonalhexagram with 6 identical vertices and 2 edge lengths.[2] |  Isogonal convexoctagon with blue and red radial lines of reflection |  Isogonal "star"tetradecagon with one vertex type, and two edge types[3] |
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| Distortedsquare tiling |
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| A distorted truncated square tiling |
Anisogonal polyhedron and 2D tiling has a single kind of vertex. Anisogonal polyhedron with all regular faces is also auniform polyhedron and can be represented by avertex configuration notation sequencing the faces around each vertex. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration.
| D3d, order 12 | Th, order 24 | Oh, order 48 | |
|---|---|---|---|
| 4.4.6 | 3.4.4.4 | 4.6.8 | 3.8.8 |
 A distortedhexagonal prism (ditrigonal trapezoprism) |  A distortedrhombicuboctahedron |  A shallowtruncated cuboctahedron |  A hyper-truncated cube |
Isogonal polyhedra and 2D tilings may be further classified:
These definitions can be extended to higher-dimensionalpolytopes andtessellations. Alluniform polytopes areisogonal, for example, theuniform 4-polytopes andconvex uniform honeycombs.
Thedual of an isogonal polytope is anisohedral figure, which is transitive on itsfacets.
A polytope or tiling may be calledk-isogonal if its vertices formk transitivity classes. A more restrictive term,k-uniform is defined as ak-isogonal figure constructed only fromregular polygons. They can be represented visually with colors by differentuniform colorings.
 Thistruncated rhombic dodecahedron is2-isogonal because it contains two transitivity classes of vertices. This polyhedron is made ofsquares and flattenedhexagons. |  Thisdemiregular tiling is also2-isogonal (and2-uniform). This tiling is made ofequilateral triangle and regularhexagonal faces. |  2-isogonal 9/4enneagram (face of thefinal stellation of the icosahedron) |