The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that a certain subset of the Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal with respect to the Z^2 action of translation. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of m × n configurations. / Graduate / 0405