Tuneable shapes and uniformity of the laser-induced periodic surface structures (LIPSS) attract interest because of their diverse applications in both scientific research and technological advancements. In this work, we investigate the progression of regular one-dimensional (1D) LIPSS on a tungsten surface, examining its evolution based on the time delay between two laser pulses that initiate the formation of nanoripples. 1D-LIPSS were formed in the case of single-beam laser ablation with approximately 84 laser pulses. Two-dimensional (2D) LIPSS, including triangle, hexagonal, and square shapes, were generated by employing two cross-polarized laser beams (with wavelengths of 1030 nm and pulse durations of 40 fs) with no delay between the pulses. However, introducing a time delay of 2 picoseconds (ps) between the two cross-polarized laser pulses resulted in the division of the initially cross-oriented 2D-LIPSS into square-shaped structures, particularly along the spatially overlapped region of the laser beams. The emergence of triangle- and hexagonal-shaped two-dimensional laser-induced periodic surface structures (2D-LIPSS) is analyzed within the framework of a self-organization model, particularly in the context of solidifying the molten layer under cold temperature conditions on the tungsten surface. We examine the formation mechanism of LIPSS, attributing it to a combination of the self-organization/hydrodynamic mechanism initially, which transitions to the electromagnetic mechanism as the effective number of pulses increases.