I'm study the Bruhat-Tits building of $PGL_n(K)$, where $K$ is a complete local field. In particular, im tried to understand the geometric properties of the apartment in special case $n=4$, which are tilings of Euclidean space $\mathbb{R}^{3}$. In the case $n=3$, the tiling (apartments) of the plane is generated by equilateral triangles. But in the case $n=4$ the tiling cannot be generated by regular tetrahedrons. So, which are the "good" tetrahedrons that tiling the space?
Im guess that the tetrahedrons have dihedral angles $\frac{\pi}{2}$, $\frac{\pi}{3}$. The angles are obtained by the Coxeter matrix $(m_{i,j})$ related with the Weyl group $W$ generated by reflections on the faces of fundamental tetrahedron, but im not sure of that.
Thanks.