Rank

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. Specifically, polygons have rank 2, polyhedra have rank 3, polychora have rank 4, and so on. 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.

Idea

Rank is a combinatorial property of a polytope, meaning it is based on how the relationship between elements of the polytope, rather than their locations in space. Rank classifies the difference between e.g. polygons and polyhedra as being about the elements that comprise them rather than about what space they are in.

A rank n  polytope is composed of rank n-1  polytopes as its facets; with the point being rank 0.

Definition

Abstract polytope

Formally, the rank of a polytope is defined as the common length of all flags, minus 2.

Since each flag contains 2 improper elements, subtracting 2 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.

Distinguished generators

In terms of distinguished generators, the rank of a polytope is the cardinality of the set of generators.

Rank and dimension

The cubic honeycomb is a polychoron (rank 4), but its vertices span 3-dimensional space.

The blended hexagon is a polygon (rank 2), but its vertices span 3-dimensional space.

The rank of a polytope and the dimension of its space are often coincident. For example a cube is rank 3 and is realized in 3-dimensional space. This similarity means that the terms are used somewhat interchangeably.

However there are cases where the dimension and rank are not equal.

• Tessellations are usually realized in a space with dimension one less than their rank. For example the triangular tiling has vertices in 2-dimensional space but it is rank 3.
• Ditopes are made of two perfectly coincident facets, and thus when realized they have dimension equal to that of their facets. For example the square dihedron is rank 3, but it lies in the same 2-dimensional space as a square.
• Many skew polytopes are realized in space whose dimension exceeds their rank. For example the blended octahedron is a polyhedron (rank 3), but it vertices span a 4-dimensional space.

In the context of skew polytopes in Euclidean space, the dimension of a polytope can be defined as the dimension of affine span of its vertices. In other words, the lowest number of dimensions required for an affine subspace to enclose all the vertices.

Bibliography

• Fernandes, Maria (2014). "Regular and chiral hypertopes" (PDF).