The lab
Guilloche
Guilloché patterns are the intricate geometric engravings found on banknotes, stock certificates, fine watches, and passport pages. They are generated by combining sinusoidal waves at different frequencies into parametric curves that loop and overlap to create dense, mesmerizing lacework. Historically produced by rose engines — ornamental lathes with eccentric cams — they are now one of the primary anti-counterfeiting features in security printing.
r(t) = R + A·sin(n₁·t + φ₁) + B·sin(n₂·t + φ₂) · x = r·cos(t), y = r·sin(t)
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The word guilloché comes from the French guillocher, meaning to engrave with intersecting curved lines. The technique dates to at least the 16th century, when ornamental lathes called "rose engines" were used to cut decorative patterns into metal, ivory, and wood. The operator would press a graving tool against a spinning workpiece whose axis was gently displaced by an eccentric cam, producing spirographic waves that overlapped into fantastically dense webs. The resulting patterns were prized both for their beauty and for their near-impossibility to reproduce by hand.
By the 19th century, guilloché had become the backbone of security printing. The U.S. Bureau of Engraving and Printing, the Bank of England, and their counterparts worldwide adopted increasingly complex rose-engine patterns for banknotes and bonds. The logic is simple: if a pattern requires a precision machine to produce, then any copy made without that machine will betray itself. Even today, when most counterfeiting is digital, the fine line structures of guilloché remain difficult for consumer printers to resolve.
This lab generates the curves mathematically. The parametric formula modulates a base radius with two sinusoidal components of different frequencies, producing Lissajous-like interference when projected around a circle. Varying the wave counts, amplitudes, and phases produces an enormous space of unique patterns — try the randomize button to explore it.