A flag of an 
-dimensional polytope is a
nested collection of nonempty proper faces,
one of each dimension 0, 1, ..., 
,
with each contained in the next. The empty face and the
whole polytope are sometimes included as well (McMullen
and Schulte 2002, p. 11). In a simplicial
complex, flags are nested collections of simplices.
For a three-dimensional polytope, a flag consists of a vertex, an edge containing that vertex, and a face containing that edge.
