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 c ...

, 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 t ...

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

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

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 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 In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

''. 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, 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 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 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 polygons are ''isogonal''. The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of an isogonal polygon is an

isotoxal polygon In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...

. 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 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 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 The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

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

isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...

(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

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

(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

s and tessellations. All uniform polytopes are ''isogonal'', for example, the uniform 4-polytopes 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 t ...

s. The

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 congru ...

, 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

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 Vladimir may refer to: Names * Vladimir (name) for the Bulgarian, Croatian, Czech, Macedonian, Romanian, Russian, Serbian, Slovak and Slovenian spellings of a Slavic name * Uladzimir for the Belarusian version of the name * Volodymyr for the Ukra ...