Abstract: Given a reduced crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$ and a polytope $P$ which is the convex hull of the $W$-orbit of a dominant weight. The polytope $P$ is associated to a real toric varieties $X_P^{\mathbb{R}}$. We will see that the underlying topological space $X_P^{\mathbb{R}}/W$ is contractible when the rank is below 7.