Bank notes (and other certificates) typically have patterns of fine lines, for example the pattern behind the lion here, from the Bank of England Museum:
Do these patterns have a name? What is the proper name for these patterns?
You're looking for the term Guilloché pattern:
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. — Wolfram MathWorld
More examples at Math Games.
More informally, they're known as Spirograph patterns. A Spirograph is a popular toy so this is the most well-known name.
Barmar asked a follow up question on Money SE about how this works: How does Guilloché pattern inhibit counterfeiting?