FareModular

FareModular is an educational iOS application used to explore the geometric shapes of the upper half plane, Farey diagrams, and actions of the modular group PSL (2, T). It is built using UIKit for iOS 14.0 and higher versions, and can run on iPhone and iPad with adaptive layout that allows for clear scaling on screen size. This application visualizes how rational vertices divide boundaries at infinity through Farey sequences and their hyperbolic geodesic connections. You can check the standard fundamental domain, observe the tiling plane of hyperbolic triangles under SL (2, T) transformation, and interactively apply generators T (z ↦ z+1) and S (z ↦ − 1/z) to track modular trajectories. A dedicated module connects these images to the module form, displaying how the position of the vertices is related to the expansion of q, and how the weights affect the behavior of the Einstein series. The interactive controls on each screen include** UIStep**、UISwitch、UI SegmentedControl、UITextField、UICutton And * * UIPickerView * * - connected to real-time canvas updates on the upper half plane. You can type complex numbers and SL (2, T) matrices, select Farey score pairs, gradually build generator chains, and read hyperbolic distances along geodesics. The main screen features a fun cartoon calf farm theme; Each laboratory screen uses a card based control panel with single space readings for coefficients and reduced trajectories.