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Fractal geometries have been studied for antennas with special characteristics, both as antenna elements, and as spatial distribution functions for elements in antenna arrays. Koch dipole and monopole antennas have been studied recently as smaller sized antennas, and their multi-band characteristics have been reported (see Cohen, N., IEEE Professional Program Proc. of Electronics Industries Forum of New England, p.43-9, 1997; Puente-Baliarda, C., IEEE Trans. Antennas Propagat., vol.48, p.1773-81, 2000). We try to establish a link between the interval of multi-band characteristics of dipole Koch curve antennas with the dimension of the underlying fractal geometry.