the Euclidean plane by regular triangles. The corresponding tiling group is the Coxeter group of affine type $\widetilde{{\rm A}_2}$; it is generated by the reflections in the three lines supporting the edges of any fundamental triangle.

Note that Poincaré’s theorem is a concrete source of Euclidean tilings: start with a Euclidean polyhedron in which each dihedral angle between codimension 1 faces is of the form ${\pi \over m}$ for some integer $m \geqslant 1$ (depending on the pair of faces), then the group generated by the reflections in these faces is an affine Coxeter group [@Maskit IV.H.11].

In what follows, $\Sigma$ is a Euclidean tiling giving rise to a Euclidean reflection group by Poincaré’s theorem (in Bourbaki’s terminology, it can also be seen as the natural geometric realization of the Coxeter complex of an affine Coxeter group, that is the affinization of the Tits’ cone of the latter group [@Lie456]).

\[defi - simplicial building\] Let $(\Sigma, W)$ be a Euclidean tiling and its associated Euclidean reflection group. A [(discrete) Euclidean builiding]{} of type $(\Sigma, W)$ is a polysimplicial complex, say $\mathcal{B}$, which is covered by subcomplexes all isomorphic to $\Sigma$ – called the [apartments]{} – such that the following incidence 