geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, 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 that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

) or a

tiling Tiling may refer to: *The physical act of laying tiles * Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester T ...

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 The face is the front of an animal's 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 aff ...

in the same or reverse order, and with the same

angles The Angles ( ang, Ængle, ; la, Angli) were one of the main Germanic peoples who settled in Great Britain in the post-Roman period. They founded several kingdoms of the Heptarchy in Anglo-Saxon England. Their name is the root of the name ...

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 In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

. The

pseudorhombicuboctahedron In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same patt ...

which 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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

, apeirogons and

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

s 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 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 faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

and can be represented by a

vertex configuration In geometry, a vertex configurationCrystallography ...

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 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 faces are the same. More specifically, all faces must be not merely congruent ...

(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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

. * '' Quasi-regular'' if it is also isotoxal (edge-transitive) but not

(face-transitive). * '' Semi-regular'' if every face is a regular polygon but it is not

(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 clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...

'' 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

(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-dimensional ...

. All

uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...

s 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 polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. Th ...

s and

convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar cubic honeycomb and 7 tr ...

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 faces are the same. More specifically, all faces must be not merely congrue ...

, which is transitive on its

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

''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

s. They can be represented visually with colors by different

uniform coloring In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following differ ...

References

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

External links