Guilloche patterns are critically important in banknote designs. They are created mechanically with a geometric lathe, or more likely these days, mathematically, such as this “Guilloche leg”, which is created with the equation next to it:
When creating, or even looking at, a banknote design, one of the first things you realize is their inherent and very deliberate imperfections. There will be an apparent mis-registration of colour, a strangely ragged line, a discontinuity in a pattern or an odd serif or ligature on a piece of lettering, but it�s exactly how it was designed. Without it, it wouldn�t be right. The design of banknotes represents imperfect perfection � if it was perfect by our usual standards, it would be imperfect.
The geometric or �looping line patterns� you see in the border work of Federal Reserve Notes are commonly referred to as Guilloche patterns. Engravers originally used a mechanical lathe machine (also called a geometric lathe) to produce patterns of varying sizes and shapes depending on the needs of the banknote design. Imagine this machine as being a �super Spiro-graph�. The machine would be set up using a series of precision gears, which would direct a stylus to lightly cut the geometric pattern into a steel Die to create the pattern when the paper money was printed. Today software is used to create new and more complex patterns as well as digitally capturing archived images to design or incorporate the classic guilloches originally produced by the geometric lathe.