Given a simple connected compact Lie group $K$ and a maximal torus $T$ of $K$, the Weyl group $W=N_K(T)/T$ naturally acts on $T$. First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit $W$-equivariant triangulation of $T$. We describe the associated cellular homology chain complex and give a formula for the cup product on its dual cochain complex, making it a $\mathbb{Z}[W]$-dg-algebra. Next, remarking that the combinatorics of this dg-algebra is still valid for Coxeter groups, we associate a closed compact manifold $\mathbf{T}(W)$ to any finite irreducible Coxeter group $W$, which coincides with a torus if $W$ is a Weyl group and is hyperbolic in other cases. Of course, we focus our study on non-crystallographic groups, which are $I_2(m)$ with $m=5$ or $m\ge 7$, $H_3$ and $H_4$. The manifold $\mathbf{T}(W)$ comes with a $W$-action and an equivariant triangulation, whose related $\mathbb{Z}[W]$-dg-algebra is the one mentioned above. We finish by computing the homology of $\mathbf{T}(W)$, as a representation of $W$.