Integral quadratic forms q : Z n → Z , with n ≥ 1 , and the sets R q ( d ) = { v ∈ Z n ; q ( v ) = d } , with d ∈ Z , of their integral roots are studied by means of mesh translation quivers defined by Z -bilinear morsifications b A : Z n × Z n → Z of q , with Z -regular matrices A ∈ M n ( Z ) . Mesh geometries of roots of positive definite quadratic forms q : Z n → Z are studied in connection with root mesh quivers of forms associated to Dynkin diagrams A n , D n , E 6 , E 7 , E 8 and the Auslander–Reiten quivers of the derived category D b ( R ) of path algebras R = K Q of Dynkin quivers Q . We introduce the concepts of a Z -morsification b A of a quadratic form q , a weighted Φ A -mesh of vectors in Z n , and a weighted Φ A -mesh orbit translation quiver Γ ( R q , Φ A ) of vectors in Z n , where R q ≔ R q ( 1 ) and Φ A : Z n → Z n is the Coxeter isomorphism defined by A . The existence of mesh geometries on R q is discussed. It is shown that, under some assumptions on the morsification b A : Z n × Z n → Z , the set R q ∪ Ker q admit a Φ A -orbit mesh quiver Γ ( R q ∪ Ker q , Φ A ) , where Φ A : Z n → Z n is the Coxeter isomorphism defined by A . Moreover, Γ ( R q ∪ Ker q , Φ A ) splits into three infinite connected components Γ ( ∂ A − R q ) , Γ ( ∂ A + R q ) , and Γ ( ∂ A 0 R q ∪ Ker q ) , where Γ ( ∂ A − R q ) ≅ Γ ( ∂ A + R q ) are isomorphic to a translation quiver Z ⋅ Δ , with Δ an extended Dynkin quiver, and Γ ( ∂ A 0 R q ∪ Ker q ) has the shape of a sand–glass tube.