The apparatus includes a plurality of movable support members that are arranged side-by-side and collectively form a quadratic carrier surface, such as a cylindrical carrier surface. A plurality of fiber strand carrier members are movable on the carrier surface to intertwine the fiber strands. Each fiber strand extends from its carrier member along an axis that intersects the axes of the other fiber strands at a generally common consolidation point near a fiber gathering device that gathers the fiber strands from the carrier members. The carrier members are generally equidistant from the fiber consolidation point as a result of moving on the quadratic carrier surface so that the gathering device can maintain a generally constant tension in the fiber strands regardless of their position on the carrier surface.