Modular Forms — Fundamental Domain Explorer

This simulator visualizes the action of the modular group SL(2,Z) on the upper half-plane. The classic fundamental domain of the group — the hyperbolic "triangle" bounded by |z| ≥ 1 and |Re(z)| ≤ ½ — is drawn in the canvas above, surrounded by neighboring copies produced by the generators T (translation) and S (inversion). Click anywhere to drop a point and use the T/S buttons to move it through the tiling, watching its orbit connect matching points across different fundamental-domain copies. The strip below plots a truncated Eisenstein series E₄(z) sampled live along the boundary of the fundamental domain, a first taste of how modular forms behave along that same contour.