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Where do the geometries come from? I've been looking at embedded PCB antennas and also modules that attach to PCBs and they always have exotic looking notches and curves. Some are 3D while others are flat 2D styles. How does one actually arrive at such geometries mathematically? They seem so random and I know that their are exact subtleties to why the specific combination of geometries are in place, and even then, these geometries depend on the materials being used.

In the cellular industry, is it mostly intuitive knowledge combined with an iterative design approach that these designers come to their final antenna design, or do they use special design tools/simulations or mathematics/physics theory to get there? I know its probably a combination, but I'm wondering what it takes to actually design an antenna.

I'm particularly interested in the design of cellular quad band passive antennas, so if anyone has any experience or suggestions on where to start, I would appreciate it greatly.