Given a simple connected compact Lie group K and a maximal torus T of K, the Weyl group W=NK(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 W-dg-ring.For a non-crystallographic Coxeter group W, using compact hyperbolic extensions rather than affine ones, we construct a compact W-manifold T(W), which is an analogue of a torus for W. We exhibit a W-equivariant triangulation of T(W) and compute the associated W-dg-ring. Also, we derive its homology representation.