The heat map (using Eq. 2.1) for the resulting configuration space in ℝ 2 is shown in Fig, . Examination of the space reveals that the configurations are periodic in two directions. This is indicated by the grey dotted grid in Fig. and an example of a single representative injective mapping is indicated by the white dashed rhombus. This doubly periodic property can be exploited to transform a single injective mapping in ℝ 2 → 𝑇 2 , as shown in Fig. , representing a form of compactification. Linear trajectories through the AB3 ETP-3 E′-mode configuration space, for example the green solid line shown in Fig. , manifest as paths in the 𝑇 2 representation when periodic replication in the ℝ 2 representation 𝒞 3E ′ (green dotted lines in Fig. ) is included. If the gradient of the trajectory in ℝ 2relative to the 1: 2 √3 ⁄ proportions of the injective mappingis rational, then the trajectory in the 𝑇 2 representation will be a closed path. If the gradient is irrational, then the trajectory will densely cover the 𝑇 2 space.