 Winding around the circle requires energy, because the string must be stretched against its tension, so it contributes an amount of energy of the form formula_1, where formula_2 is a constant called the "string length" and w is the winding number (an integer).Now (for a given maximum energy) there will be many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R. In fact, a theory with large R and a theory with small R are equivalent, where the role of momentum in the first is played by the winding in the second, and vice versa.Mathematically, taking R to formula_3 and switching n and w will yield the same equations.