Those three-color roundels are more accurately known as 'cockades'. They go back as a distinguishing insignia to well before the American Revolution, when designs made of cloth or silk would be pinned onto the side of a cap (or painted on a leather or brass helmet), usually on the left, to signify the soldiers national or political alliance. They continued in use well into the 20th century, especially in Germany. Those red-black-white roundels on the sides of Pickelhauben and the front of field caps are an evolution of the same idea.
The simple black cockade seen above was the American design (at least among the Continentals) until the alliance with the French was announced, when it was ordered that a white center would be added to the cockade to commemorate it.
Before-
After-
You can also see the German ones on these two below-
(where the chin strap attaches to the Pickelhaube)
(and on the front of the field cap)
And, of course, still in use for that in WWII-
