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This paper relates for the first time, multiple resonant frequencies of fractal element antennas using Koch curves to their fractal dimension. Dipole and monopole antennas based fractal Koch curves studied so far have generally been limited to certain standard configurations of the geometry. It is possible to generalize the geometry by changing its indentation angle, to vary its fractal similarity dimension. This variation results in self-similar geometry which can be generated by a recursive algorithm. Such a variation is found to have a direct influence on the input characteristics of dipole antennas. The primary resonant frequency, the input resistance at this resonance, and the ratio of first two resonant frequencies, have all been directly related to the fractal dimension. Curve-fit expressions can also be obtained for the performance of antennas at their primary resonance, in terms of fractal iteration and fractal dimension. The antenna characteristics have been studied using extensive numerical simulations and are experimentally verified. These findings underscore the significance of fractal dimension as an important mathematical property of fractals that can be used as a design parameter for antennas. The use of these ideas would not only reduce the computational intensity of optimization approaches for design of fractal shaped antennas, but also help antenna designers approach the problem systematically. Design formulation for antennas based on other fractal geometries can be similarly obtained after identifying suitable parameters of variation. This would therefore help analytical design of multiband and multifunctional antennas using fractal geometries.