Flag graphs

The reduced flag graph of a polytope is useful for understanding the combinatorial structure of the polytope, as well as relating the structures of different polytopes. These can be considered voltage graphs in the sense of maniplexes (see Hubard, Mochán, and Montero).

An abstract polytope has a set of flags. Flags can be connected by an operation (the flag change) that replaces the rank-k element of the flag with another element while fixing the rest, which by the diamond property is unique. This results in the flag graph of the polytope.

A tetrahedron

Here is the flag graph of a tetrahedron. The color of the edges and the number of tick marks per edge refer to the rank of the element that is being changed: gray for 0, red for 1, and gold for 2. This flag graph is very complicated and has a high degree of symmetry. By identifying flags in the same orbit under the symmetry group, we obtain the following diagram.

A symmetry-reduced tetrahedron

The self-loops indicate that performing a flag change on a flag from this orbit results in a flag from the same orbit. Because every flag has to have one edge for each possible rank, the self-loops are superfluous and their presence will only be implied on subsequent diagrams.

A symmetry-reduced tetrahedron with self-loops removed

Uniform polytopes

Two dimensions

The simplest kind of graph is one where all flags are in the same orbit: a regular polytope. This graph looks the same regardless of dimension.

A regular polytope
In two dimensions, the regular polygon is the only kind of uniform polytope. However, in addition to being uniform with its full symmetry group, it can also be considered uniform with its chiral subsymmetry, and even-sided polygons can also be considered uniform as truncates. These can be helpful when analyzing higher-dimensional uniform polyhedra.
A chiral polygon
A truncated polygon

Three dimensions

The reduced flag graphs for the Archimedean solids are shown below.

A rectified regular polyhedron
A truncated regular polyhedron
An expanded regular polyhedron
An omnitruncated regular polyhedron
A snub regular polyhedron

The reduced flag graphs of all uniform polyhedra can be found on the uniform polyhedra page.

Four dimensions

The reduced flag graphs of many uniform polychora can be found on the uniform polychora page.

Five dimensions

The reduced flag graphs of some uniform polytera can be found on the uniform polytera page.