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While most of what I've read about fractals has been dubious in nature, over the years, I keep hearing that these sorts of self-similar (or approximately self-similar) geometries are useful in the manufacture of high-performance antennas at small scales, perhaps for cell phones or distributed sensing applications. Benoit Mandlebrot himself cited this as one application of his work.

Can anyone provide me an intuitive reason why a fractal geometry, as opposed to some other symmetric structure, would be optimal for compact antenna design?

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Fractal structures often are symmetric structures. The symmetry is a scaling symmetry. –  Raskolnikov Aug 25 '11 at 11:29
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@Raskolnikov, why is scaling symmetry important? –  TheSheepMan Aug 25 '11 at 12:39
    
Scaling symmetry is useful for an antenna if you want it to be broadband (i.e. you want it to work well over a large range of wavelengths), which you typically do. –  user1631 Aug 25 '11 at 17:34
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The presence of scaling symmetry means that the impedance of the antenna can be made roughly the same across a large range of frequencies. This allows you to build wide bandwidth antennae. –  Willie Wong Aug 25 '11 at 17:35
    
@Raskolnikov: could one of you answer the question? It's a good question. –  Ron Maimon Sep 4 '11 at 21:56

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One advantage of fractal antennae is their larger bandwidth, which is good because it allows the same antenna to access more frequency bands, and the use of larger bandwidth for frequency modulated signals allows for also larger data throughput.

An imprecise and not-entirely-correct-in-the-details-of-electrical-engineering explanation for this increased bandwidth is that the presence of scaling symmetry means that the impedance of the antenna can be made roughly the same across a large range of frequencies, since the impedance depends on the difference between the resonant frequency and the signal frequency, and the resonant frequency depends on the size of characteristic features in the antenna.

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