# Transitivity

A certain set of elements in a polytope are **transitive** when any element in that set can be turned into any other element in that set through any amount of isometries. Usually, the "set" of elements are all the elements of a certain rank.

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**: the prefix -gon refers to points, not lines, for historical reasons. The prefix -tox is used for edge transitivity instead) 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 (isotopal)*, and those with transitive *flags* are *flag transitive*. Polytopes where all sets of elements by rank are transitive are *fully transitive*. Polytopes that are both isogonal and isotopal are known as **noble polytopes**.

All isotopic and isogonal polytopes are orbiform.