This page illustrates the effect of transformations in the complex plane. The grey canvas shows the upper half-plane (real part from -1 to +1, and imaginary part > 0). Everything is driven by the location z of the mouse pointer, with three transformations: S: z → -1/z, T: z → z+1, and U: z → z-1. By repeating these ad infinitum, a large set of transformations is generated, specifically the group SL2(Z). More information from Keith Conrad. In this canvas, the colour is more red when T is applied, more green when U is applied, and the blue component is on/off when the number of S transformations is odd/even. A point may have several colours if a sequence of transformations brings z back to where it started. White lines are the boundaries of the fundamental domains, where only one point can be. This group of transformations is especially beautiful in combination with complex analysis, searching for functions that are both differentiable and invariant under SL2(Z) transofrmations. For a wild ride, try these lectures.