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We created hexagonal rings on a semiconductor surface by atom manipulation in a scanning tunneling microscope (STM). Our measurements reveal the generic level structure of a quantum ring, including its single ground state and doubly degenerate excited states. The ring shape leads to a periodic potential modulation and thereby a perturbation of the level structure that can be understood in analogy to band gap formation in a one-dimensional periodic potential. The modulation can be enhanced or inverted by further adding or removing atoms with the STM tip. Our results demonstrate the possibility of designing and controlling electron dynamics in a tunable periodic potential, holding promise for the construction of two-dimensional artificial lattices on a semiconductor surface.