Rank

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All flags in a cube have 3 proper elements, so its rank is 3.

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.

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.

It is possible to define a more general notion of an abstract polytope, where instead of a partial order, one is left only with an incidence relation. These structures are known as hypertopes. Instead of ranks, their elements have types with no particular order.