For graph theory, see Vertex-transitive graph. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics, a vertex-transitive graph is a graph G such that given any two vertices v1 and v2 of G, there is some

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are the same. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions In Geometry, a vertex (plural "vertices" is a special kind of point. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces. In Geometry, a face of a Polyhedron is any of the Polygons that make up its boundaries

Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or For the Mechanical engineering and Architecture usage see Isometric projection. Other ways of saying this are that the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit.

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 Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

## Isogonal polygons

An example isogonal octagon with D4 symmetry. Regular octagons A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size

All regular polygons and regular star polygons are isogonal. General properties These properties apply to both convex and star regular polygons Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one

Some even-sided polygons which alternate two edge lengths, for example rectangle, are isogonal. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as

All such 2n-gons have dihedral symmetry (Dn, n=2,3,. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections . . ) with reflection lines across the mid-edge points.

## Isogonal polyhedra

Isogonal polyhedra may be 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. A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex General properties These properties apply to both convex and star regular polygons
• Quasi-regular if it is also isotoxal (edge-transitive) but not isohedral (face-transitive). A Polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.
• Semi-regular if every face is a regular polygon but it is not isohedral (face-transitive) or isotoxal (edge-transitive). A semiregular polyhedron is a Polyhedron with regular faces and a symmetry group which is transitive on its vertices (Definition varies among authors; e. g. some exclude solids with dihedral symmetry, or nonconvex solids. )
• Uniform if every face is a regular polygon, i. A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i e. it is regular, quasiregular or semi-regular.
• Noble if it is also isohedral (face-transitive). A noble polyhedron is one which is Isohedral (all faces the same and Isogonal (all vertices the same

An isogonal polyhedron has a single kind of vertex figure. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off If the faces are regular (and the polyhedron is thus uniform) it can be represented by a vertex configuration notation sequencing the faces around each vertex. In Polyhedral Geometry a vertex configuration is a short-hand notation for representing a polyhedron Vertex figure as the sequence of faces around a vertex

## Isogonal polytopes and tessellations

These definitions can be extended to higher dimensional polytopes and tessellations. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations In Geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps Most generally, all uniform polytopes are isogonal, for example, the uniform polychorons and convex uniform honeycombs. A uniform polytope is a Vertex-transitive Polytope made from uniform polytope facets. In Geometry, a uniform polychoron (plural uniform polychora) is a Polychoron or 4- Polytope which is Vertex-transitive In Geometry, a convex uniform honeycomb is a uniform space-filling Tessellation in three-dimensional Euclidean space with non-overlapping convex uniform

The dual of an isogonal polytope is called an isotope which is transitive on its facets. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the A facet of a Simplicial complex is a maximal simplex In the general theory of Polyhedra and Polytopes two conflicting meanings are currently jostling for acceptability

## References

• Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 9-521-55432-2, p. 369 Transitivity