Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.

Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.

The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features guilloché patterns.

REFERENCES:

Pegg, E. "Math Games: Guilloché Patterns." Feb. 9, 2004. http://www.mathpuzzle.com/MAA/13-Guilloche%20Patterns/mathgames_02_09_04.html.