we provisionally used the terminology “cells” instead of “polysimplices” to state the axioms (as already mentioned, cells will henceforth be called facets – alcoves when they are top-dimensional).

Let us provide now some examples of discrete buildings corresponding to the already mentioned examples of Euclidean tilings.

\[ex - simplicial buildings\]

-   The class of buildings of type $({\bf R}, {\rm D}_\infty)$ coincides with the class of trees without terminal vertex (recall that a tree is a $1$-dimensional simplicial complex – i.e., the geometric realization of a graph – without non-trivial loop [@SerreArbres]).

-   A $2$-dimensional $\widetilde{{\rm A}_2}$-building is already impossible to draw, but roughly speaking it can be constructed by gluing half-tilings to an initial one along [walls]{} (i.e., fixed point sets of reflections) and by iterating these gluings infinitely many times provided a prescribed “shape” of neighborhoods of vertices is respected – see Example \[ex - SL3bis\] for further details on the local description of a building in this case.

It is important to note that axiom (ii) does [*not*]{} require that the isomorphism between apartments extends to a global automorphism of the ambient building. In fact, it may very well happen that for a given Euclidean building $\mathcal{B}$ 