We have systematically studied reconstructions of the 6H SiC(0001) and (000 (1) over bar) surface under both Si rich and C rich condition using field ion-scanning tunneling microscopy(FI-STM). The sample was cleaned by in situ Si beam etching at 900-1000 degrees C. The Si rich and C rich phases were produced by annealing the sample in a Si flux and C2H2, respectively. On the (0001) surface, the as-cleaned surface showed ( root 3x root 3) reconstruction. With increasing Si concentration, (2x2). (2 root 3x6 root 3), (3x3), and (7x7) reconstructions were observed. Reaction of the Si rich phases with C2H2 molecule at 1050 degrees C resulted in the formation of C rich surface, which exhibited (2x2)/(6x6) reconstruction. On the (000 (1) over bar) surface. (2 root 3x2 root 3) reconstruction was observed after cleaning. Under Si rich condition, (2x2), (3x3), and (7x7) reconstructions were observed. Annealing the surface in C2H2 beam at 1050 degrees C led to C rich phase with(1x1) structure. Structure model For (root 3x root 3) and (2 root 3x2 root 3) reconstruction are proposed, and possible applications of using the surface reconstruction to selectively grow the polytype SIC is discussed.