geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

(e.g. a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

of the figure. This implies that each vertex is surrounded by the same kinds of

face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...

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 group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s and

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

. 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 In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

, 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 In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

, are ''isogonal''. All planar isogonal 2''n''-gons have

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

(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 In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

and can be represented by a

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

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 In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

. * '' 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 A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...

'' 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 A noble is a member of the nobility. Noble may also refer to: Places Antarctica * Noble Glacier, King George Island * Noble Nunatak, Marie Byrd Land * Noble Peak, Wiencke Island * Noble Rocks, Graham Land Australia * Noble Island, Gr ...

'' if it is also isohedral (face-transitive).

''N'' dimensions: Isogonal polytopes and tessellations

These definitions can be extended to higher-dimensional

s and

tessellations A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensiona ...

. All uniform polytopes are ''isogonal'', for example, the

uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

s and

convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells. Twenty-eight such honey ...

s. The dual of an isogonal polytope is an

isohedral figure In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...

, 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 a ''k-isogonal figure'' constructed only from

s. They can be represented visually with colors by different uniform colorings.

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) {{DEFAULTSORT:Isogonal Figure Polyhedra Polytopes

Isogonal polygons and apeirogons

Isogonal polyhedra and 2D tilings

''N'' dimensions: Isogonal polytopes and tessellations

''k''-isogonal and ''k''-uniform figures

See also

References

External links