Rank

Rank is the intrinsic property of a polytope that distinguishes polygons, polyhedra, polychora, and others. Rank is often conflated with dimension, although that can have various other different meanings. The rank of a polytope does not depend on how it is realized.

Abstract polytope
Formally, the rank of a polytope is defined as the common length of all flags, minus 2. Subtracting 2 is mostly an arbitrary convention that makes it so that rank and other notions of dimension coincide under most circumstances. For instance, a cube has rank 3, and is most naturally realized in a 3-dimensional space.

Incidence system
The rank of a incidence system is the cardinality of the type set. This gives a notion of rank to hypertopes even though their flags do not have a common length. When a abstract polytope is viewed as a hypertope the two definitions coincide.