Isogonal figure

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

Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope '' acts transitively'' on its vertices, or that the vertices lie within a single ''symmetry orbit''.

All vertices of a finite -dimensional isogonal figure exist on an -sphere.

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

The pseudorhombicuboctahedronwhich is ''not'' isogonaldemonstrates 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

All regular polygons, apeirogons and regular star polygons are ''isogonal''. The dual of an isogonal polygon is an isotoxal polygon.

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

All planar isogonal 2''n''-gons have dihedral symmetry (D_''n'', ''n'' = 2, 3, ...) with reflection lines across the mid-edge points.

Isogonal polyhedra and 2D tilings

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.

Isogonal polyhedra and 2D tilings may be further classified:

* '' Regular'' if it is also isohedral (face-transitive) and isotoxal (edge-transitive); this implies that every face is the same kind of regular polygon.
* '' Quasi-regular'' if it is also isotoxal (edge-transitive) but not isohedral (face-transitive).
* '' 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'' 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'' if it is also isohedral (face-transitive).

''N'' dimensions: Isogonal polytopes and tessellations

These definitions can be extended to higher-dimensional polytopes and tessellations. All uniform polytopes are ''isogonal'', for example, the uniform 4-polytopes and convex uniform honeycombs.

The dual of an isogonal polytope is an isohedral figure, which is transitive on its 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 an ''k-isogonal figure'' constructed only from regular polygons. They can be represented visually with colors by different uniform colorings.

See also

* Edge-transitive (Isotoxal figure)
* Face-transitive (Isohedral figure)

References

* Peter R. Cromwell, ''Polyhedra'', Cambridge University Press 1997, , p. 369 Transitivity
* (p. 33 ''k-isogonal'' tiling, p. 65 ''k-uniform tilings'')

External links

*
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, W
VRML models


* ttp://probabilitysports.com/tilings.html List of n-uniform tilings* (Also uses term k-uniform for k-isogonal)

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Polyhedra
Polytopes