Coxeter complex $(\amod,W)$ is a CAT(1) metric space $\B$ equipped with a collection of isometric embeddings $\kappa:\amod\to\B$, called [*charts*]{}. The image of a chart is an [*apartment*]{} in $\B$. One requires that any two points are contained in an apartment and that the coordinate changes between charts are induced by isometries in $W$. (The precise axioms can be found e.g. in [@qirigid] and [@habil].) We will use the notation $\angle$ for the metric on $\B$.

We assume that $W$ fixes no point, equivalently, that $\simod$ is a simplex with diameter $\leq\pihalf$.

Via the atlas of charts, the spherical building inherits from the spherical Coxeter complex a natural structure of a [*simplicial complex*]{} where the simplices are the images of the simplices in the model apartment. As already mentioned, the images of the charts are called [*apartments*]{}. Accordingly, the images of chambers (walls, half-apartments, singular spheres) in $\amod$ are called chambers (walls, half-apartments, singular spheres) in the building. The codimension one faces are called [*panels*]{}. The [*interior*]{} $\inte(\tau)$ of a simplex $\tau$ is obtained by removing all proper faces; the interiors of simplices are called [*open simplices*]{}. The simplex [*spanned*]{} by a point is the smallest simplex containing it, equivalently, the 