As seen in the Hyperbolic \(4g\)-gons section, you can simulate flows on fundamental domains by fixing a particular fundamental domain and observe the flow traveling through the domain. Another, yet equally valid, way to visualize dynamics on fundamental domains is to fix the perspective on the flow as it travels through a tiling of fundamental domains. Below is an animation of a flow traveling through a tiling of a 1-holed torus fundamental domains (1-1 squares in the Euclidean plane).

As described in the Hyperbolic \(4g\)-gons section, each point in a fundamental domain corresponds to a particular point on a torus through a continuous mapping from the fundamental domain to the torus. A fundamental domain of a \(n\)-holed torus requires a \(4n\)-sides polygon, so to visualize motion on a \(n\)-holed torus where \(n>1\), we must move to the hyperbolic plane. Below is an animation of a flow traveling through a tiling of a fundamental domain for a \(2\)-holed torus.