Sacred Geometry Coloring Book (Digital Edition)

The complex plane is the natural home of conformal transformations — maps that preserve angles locally while warping distances. Möbius transformations of the form f(z) = (az + b) / (cz + d) move circles to circles and lines to lines, and they generate every angle-preserving symmetry of the Riemann sphere. The Joukowski transform w = z + 1/z maps a circle to an airfoil cross-section — the same one used in early aerodynamics to compute lift. Apply these maps to a regular grid and the result is unmistakably geometric and unmistakably mathematical: the lines bend exactly as the complex algebra demands. Each image and animation in this collection is computed from the underlying complex function — no Bézier curves, no artistic license.

f(z) = (az+b)/(cz+d), w = z + 1/z