Transitivity

From Polytope Wiki
(Redirected from Transitive)
Jump to navigation Jump to search

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 isogonal.

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]

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

See also[edit | edit source]