In this paper, we present a flexible design of electromagnetic bandgap (EBG) structure, which is constructed based on Fractal geometry, for antenna applications. These Fractals, which are the Sierpinski triangles, are arranged to repeat each 600to introduce the hexagonal unit cells. By changing the gap between two adjacent Sierpinski triangles inside EBG unit cell, it can be introducing two EBG structuresseparately that have broadband and dual bandgap. By using the suspending microstrip method, two arrays 3×4 of EBG unit cells areutilized to investigate the bandgap of the EBG structures. The EBG operation bandwidth of the broadband structure and the dual-band structure are about 87% and 40%; 35% at the center bandgap frequencies, respectively. Moreover, a comparison between the broadband EBG and the conventional mushroom-like EBG has been done. Experimental results of the proposed design show good agreement in comparison with simulation results. Finally, the proposed EBG structures are studied as a high impedance ground plane for enhancing the radiation properties of a patch antenna.