# Two-orbit polytope

(Redirected from Two-orbit)

A two-orbit polytope is a polytope with exactly two types of flags. Two flags are of the same type if a symmetry of the polytope maps between them. Alternatively a two-orbit polytope is a polytope where the number of flags has is twice the order of its symmetry group.

All quasiregular polytopes are either regular (one-orbit) or two-orbit, and the regular ones can be made into two-orbits by two-coloring the facets in a certain way. For example the octahedron can be two-colored to make the tetratetrahedron, a two-orbit polytope. Hypercubic honeycombs can be vertex two-colored or facet two-colored to turn them into two-orbit polytopes. All 2D semi-unifom polytopes have two-orbits, and so do their duals. Not all regular polytopes can be turned into a two-orbit by coloring them, only alternatable ones or their duals can be. Additionally, any facets or vertex figures of two-orbit polytopes are also two-orbit or regular.

All demicrosses have two-orbits, and all orthoplexes and hypercubic honeycombs can be facet two-colored to make two-orbits. Their duals, hypercubes and the hypercubic honeycombs can be vertex two-colored to make two-orbits.

## Properties

The following properties are true of two-orbit polytopes:

• A two orbit polytope is either vertex-transitive or facet-transitive.
• A two orbit polytope has at most two types of any element. e.g. there are at most two types of edges or most two types of cells.
• The dual of a two-orbit polytope is also a two-orbit polytope. More generally the dual operation preserves the number of orbits in a polytope.

## List

Here is a list, including colorings

### 2D

All of the semi-uniforms, along with their duals