# Two-orbit polytope

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[edit | edit source]

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[edit | edit source]

Here is a list, including colorings

### 2D[edit | edit source]

All of the semi-uniforms, along with their duals

### 3D[edit | edit source]

- All rectified regular polyhedra except the octahedron
- The trihexagonal tiling
- The ditrigonary triangular-hemiapeirogonal tiling
- The hexagonal-hemiapeirogonal tiling
- The square-hemiapeirogonal tiling
- The triangular-hemiapeirogonal tiling
- All uniform hemipolyhedra except gidrid and gidisdrid
- All non-compound uniform polyhedra in sidtid's regiment
- The duals of the previously mentioned polyhedra
- The tetratetrahedron
- The cube vertex two-colored such that the vertices that are each color each make tetrahedra
- The hexagonal tiling tiling two-colored such that the vertices that are each color each make triangular tilings
- The square tiling tiling two-colored such that the vertices that are each color each make square tilings
- The square tiling tiling two-colored such that the color of the faces alternate around a vertex
- The triangular tiling tiling two-colored such that the color of the faces alternate around a vertex
- The noble two-orbits ditti, the square tiling stretched so that the squares become rectangles, and the rhombic tiling

### 4D[edit | edit source]

- Sidtixhi
- Dittady
- Gidtixhi
- The tesseractihemioctachoron
- The tetrahedral-octahedral honeycomb
- The cubihemisquare honeycomb
- The octahemitriangular honeycomb
- The tetrahemitriangular honeycomb
- Idhi
- The duals of the previously mentioned polyhedra
- The cubic honeycomb two-colored such that the vertices that are each color each make alternated cubic honeycombs
- The cubic honeycomb two-colored such that the color of the cells alternate around an edge
- The hexadecachoron two-colored such that the color of the cells alternate around an edge
- The tesseract vertex two-colored such that the vertices that are each color each make hexadecachora

### 5D+[edit | edit source]

- The hypercubic honeycomb two-colored such that the vertices that are each color each make alternated hypercubic honeycombs
- The hypercubic honeycomb two-colored such that the color of the facets alternate around a peak
- The orthoplex two-colored such that the color of the facets alternate around a peak
- The hypercube two-colored such that the vertices that are each color each make demicubes
- The demicross and its dual
- The extension of the square-hemiapeirogonal tiling and the cubihemisquare honeycomb and it's dual