A method and apparatus for releasably attaching flexible abrasive disks to a flat-surfaced platen that floats in three-point abrading contact with three flat-surfaced rotatable fixed-position workpiece spindles that are mounted on spherical-rotation two-piece spindle-mount devices that are attached to a nominally-flat abrading machine base. The spindle-top flat surfaces are precisely co-planar with each other. The three spindles are positioned to form a triangle of platen supports where the rotational-centers of each of the spindles are positioned at the center of the annular width of the platen abrading-surface. Flat surfaced workpieces are attached to the spindles and the rotating floating-platen abrasive surface contacts the workpieces to perform single-sided abrading on them. The disk abrasive surfaces can be re-flattened by attaching abrasive disk-type components to the three spindles that are rotated while in abrading contact with the rotating abrasive disk. There is no wear of the abrasive-disk protected platen surface.