Random Guilloché

The Art on Every Banknote

Guilloche patterns originated in the 16th century with rose engine lathes: mechanical devices that carved interlocking curves into metal plates. The technique reached its security-printing form when Jacob Perkins used it for the first machine-printed banknotes in the 1810s. The mathematical precision of the interlocking lines made hand reproduction effectively impossible, establishing guilloches as the foundation of anti-counterfeiting technology that persists on currency, passports, and certificates worldwide.

The Mathematics

Each curve in this generator follows a parametric equation with three frequency components. The position at parameter t is: x(t) = A·cos(f₁·t + φ) + B·cos(f₂·t + φ·k) + C·cos(f₃·t), with an analogous equation for y using sine. The ratio between f₁ and f₂ determines the overall shape. Integer ratios create closed curves. The mesh texture emerges from drawing dozens of these curves with incrementally shifted phase offset φ, so each curve weaves slightly differently through the same space.

Why They Resist Forgery

A guilloche encodes its parameters implicitly. Changing any single value by even a fraction produces a visibly different pattern. A forger must reverse-engineer the exact frequencies, amplitudes, phase relationships, and line counts from the printed output alone. With dozens of overlapping semi-transparent curves, the visual complexity far exceeds what scanning and reprinting can faithfully reproduce. The gaps between lines shift at every intersection, creating a texture that photocopiers render as solid blobs rather than distinct threads.

From Lathes to Algorithms

Physical rose engine lathes used interlocking gears to convert rotary motion into complex curves. The gear ratios determined the frequency relationships. This tool replaces mechanical gears with trigonometric functions and randomized parameters. The principle is identical: precise mathematical relationships create patterns that are easy to generate, beautiful to observe, and extremely difficult to reproduce without the exact recipe. The "recipe" shown below each pattern is the digital equivalent of a lathe's gear configuration.

The Entropy Source

All pattern parameters are selected using crypto.getRandomValues(): frequency pairs, amplitudes, curve counts, loop counts, phase scaling, and palette selection. The parameter space is vast enough that generating the same pattern twice is astronomically unlikely. Each pattern is computed entirely in your browser and never transmitted to any server.