Editing Building (mathematics) (section)

==Elementary properties==
Every apartment {{mvar|A}} in a building is a [[Coxeter complex]]. In fact, for every two {{mvar|n}}-simplices intersecting in an {{math|(''n'' – 1)}}-simplex or ''panel'', there is a unique period two simplicial automorphism of {{mvar|A}}, called a ''reflection'', carrying one {{mvar|n}}-simplex onto the other and fixing their common points. These reflections generate a [[Coxeter group]] {{mvar|W}}, called the [[Weyl group]] of {{mvar|A}}, and the simplicial complex {{mvar|A}} corresponds to the standard geometric realization of {{mvar|W}}. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in {{mvar|A}}. Since the apartment {{mvar|A}} is determined up to isomorphism by the building, the same is true of any two simplices in {{mvar|X}} lying in some common apartment {{mvar|A}}. When {{mvar|W}} is finite, the building is said to be '''spherical'''. When it is an [[affine Weyl group]], the building is said to be '''affine''' or '''Euclidean'''.

The '''chamber system''' is the adjacency graph formed by the chambers; each pair of adjacent chambers can in addition be labelled by one of the standard
generators of the Coxeter group (see {{harvnb|Tits|1981}}).

Every building has a canonical [[intrinsic metric|length metric]] inherited from the geometric realisation obtained by identifying the vertices with an [[orthonormal basis]] of a [[Hilbert space]]. For affine buildings, this metric satisfies the [[CAT(k) space|{{math|CAT(0)}}]] comparison inequality of [[Aleksandr Danilovich Aleksandrov|Alexandrov]], known in this setting as the Bruhat–Tits ''non-positive curvature condition'' for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding Euclidean triangle with the same side-lengths (see {{harvnb|Bruhat|Tits|1972}}).