properties hold.

1.  Any two cells of $\mathcal{B}$ lie in some apartment.

2.  Given any two apartments, there is an isomorphism between them fixing their intersection in $\mathcal{B}$.

The cells in this context are called [facets]{} and the group $W$ is called the [Weyl group]{} of the building $\mathcal{B}$. The facets of maximal dimension are called [alcoves]{}.

The axioms of a Euclidean building can be motivated by metric reasons. Indeed, once the choice of a $W$-invariant Euclidean metric on $\Sigma$ has been made, there is a natural way the define a distance on the whole building: given any two points $x$ and $x'$ in $\mathcal{B}$, by [(SEB 1)]{} pick an apartment $\mathbb{A}$ containing them and consider the distance between $x$ and $x'$ taken in $\mathbb{A}$; then [(SEB 2)]{} implies that the so–obtained non-negative number doesn’t depend on the choice of $\mathbb{A}$. It requires further work to check that one defines in this way a distance on the building (i.e., to check that the triangle inequality holds [@Parreau Prop. II.1.3]).

\[rk - metric motivation\] The terminology “polysimplicial” refers to the fact that a building can be a direct product of simplicial complexes rather than merely a simplicial complex; this is why 