Abstract: Embodiments of the invention comprise a machine adapted for polishing work pieces such as large silicon wafers. A wafer polishing machine in accord with the invention comprises a rotatable platen in a table base, above which is mounted a lid having a head moving assembly with four synchronously rotatable head assemblies. A motor and linkage connected to the head moving assembly imparts reciprocating linear motion to the head assemblies in a selected direction in a plane parallel to an upper surface of the platen. Embodiments of the invention produce a complex relative motion between a surface of a wafer to be polished and the platen. The complex relative motion, resulting from a combination of motions including rotation of the platen, rotation of the head assemblies, and translation of the head moving assembly, improves a uniformity of polish and a rate of polishing compared to wafer polishing machines known in the art.