The control of spin waves in periodic magnetic structures has facilitated the realization of many functional magnonic devices, such as band stop filters and magnonic transistors, where the geometry of the crystal structure plays an important role. Here, we report on the magnetostatic mode formation in an artificial magnetic structure, going beyond the crystal geometry to a fractal structure, where the mode formation is related to the geometric scaling of the fractal structure. Specifically, the precessional dynamics was measured in samples with structures going from simple geometric structures toward a Sierpinski carpet and a Sierpinski triangle. The experimentally observed evolution of the precessional motion could be linked to the progression in the geometric structures that results in a modification of the demagnetizing field. Furthermore, we have found sets of modes at the ferromagnetic resonance frequency that form a scaled spatial distribution following the geometric scaling. Based on this, we have determined the two conditions for such mode formation to occur. One condition is that the associated magnetic boundaries must scale accordingly, and the other condition is that the region where the mode occurs must not coincide with the regions for the edge modes. This established relationship between the fractal geometry and the mode formation in magnetic fractals provides guiding principles for their use in magnonics applications.