We construct a geometric model for the root category \(\mathcal {D}_\infty (Q)/[2]\) of any Dynkin quiver Q, which is an \(h_Q\) -gon \(\textbf{V}_Q\) with cores, where \(h_Q\) is the Coxeter number and \(\mathcal {D}_\infty (Q)=\mathcal {D}^b(Q)\) is the bounded derived category associated to Q. As an application, we describe all spaces \(\textrm{ToSt}\mathcal {D}\) of total stability conditions on triangulated categories \(\mathcal {D}\) , where \(\mathcal {D}\) must be of the form \(\mathcal {D}_\infty (Q)\) . More precisely, we prove that \(\textrm{ToSt}\mathcal {D}_\infty (Q)/[2]\) is isomorphic to a suitable moduli space of stable \(h_Q\) -gons of type Q. In particular, an \(h_Q\) -gon \(\textbf{V}\) of type \(D_n\) is a (centrally) symmetric doubly punctured \(2(n-1)\) -gon. \(\textbf{V}\) is stable if it is positively convex and the punctures are inside the level- \((n-2)\) diagonal-gon. Another interesting case is \(E_6\) , where the (stable) \(h_Q\) -gon (dodecagon) can be realized as a pair of planar tiling pattern.
