# Transitivity

A set of elements of the same rank in a polytope is **transitive** when for any two members of the set there exists a symmetry on the polytope that can map between the two members of the set. For most purposes, a maximal set of elements of the same rank in a polytope is used.

Transitivities for elements of a certain rank are named by the name of the element followed by "-transitive" or by the elements' corresponding suffix preceded by "iso"^{[note 1]} For example, a polytope whose *vertices* are congruent under its symmetry would be *vertex*-transitive, or iso*gonal*.

If a polytope is r -transitive, where r is the rank of the transitive elements, its dual will be d-r transitive, where d is the rank of the polytope itself. For example, the duals of isogonal polyhedra are isohedral.

Other names are used for different sets of transitive elements. Polytopes with transitive *facets* are *facet-transitive* or *isotopic*, and those with transitive *flags* are *flag-transitive* or *regular*. Polytopes where all sets of elements by rank are transitive are *fully transitive* or weakly regular. *Flag transitivity* implies *full transitivity.* Polytopes that are both isogonal and isotopic are known as *noble polytopes*.

## Notes[edit | edit source]

- ↑ The suffix
*-gon*refers to points, not lines, for historical reasons. The suffix*-tox*is used for edge transitivity instead