page_title: Flag (geometry)
hierarchical_section_title: Flag (geometry)
caption_attribution_description: Face diagram of a square pyramid showing one of its flags
context_page_description: In geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.
  More formally, a flag ψ of an n-polytope is a set {F−1, F₀, ..., Fₙ} such that Fᵢ ≤ Fᵢ₊₁ and there is precisely one Fᵢ in ψ for each i. Since, however, the minimal face F−1 and the maximal face Fₙ must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces.
  For example, a flag of a polyhedron comprises one vertex, one edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces.
  A polytope may be regarded as regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.
context_section_description: In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension. 
  More formally, a flag ψ of an n-polytope is a set {F−1,  F₀, ..., Fₙ} such that Fᵢ ≤ Fᵢ₊₁ (−1 ≤ i ≤ n − 1) and there is precisely one Fᵢ in ψ for each i, (−1 ≤ i ≤ n).  Since, however, the minimal face F−1 and the maximal face Fₙ must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces.
  For example, a flag of a polyhedron comprises one vertex, one edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces.
  A polytope may be regarded as regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.
caption: Face diagram of a square pyramid showing one of its flags
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