Editing Isogonal figure (section)

{{Short description|Polytope or tiling whose vertices are identical}}
{{For|[[graph theory]]|vertex-transitive graph}}

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

Technically, one says that for any two vertices there exists a symmetry of the polytope [[Map (mathematics)|mapping]] the first [[isometry|isometrically]] onto the second. Other ways of saying this are that the [[automorphism group|group of automorphisms]] of the polytope ''[[Group action#Remarkable properties of actions|acts transitively]]'' on its vertices, or that the vertices lie within a single ''[[symmetry orbit]]''.

All vertices of a finite {{mvar|n}}-dimensional isogonal figure exist on an [[n-sphere|{{math|(''n''−1)}}-sphere]].<ref>{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.1007/PL00009307
 | issue = 1
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1453440
 | pages = 13–52
 | title = Isogonal prismatoids
 | volume = 18
 | year = 1997}}</ref>

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

The [[Elongated square gyrobicupola|pseudorhombicuboctahedron]]{{snd}}which is ''not'' isogonal{{snd}}demonstrates 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.