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A DIY fractal antenna project is described at http://www.htpc-diy.com/2012/04/diy-flexible-fractal-window-hdtv.html

Mainly idle curiosity, but I was wondering what the principles behind these might be, and if there was an optimal shape for them.

I'm guessing the idea is that self-similarity, up to a point, optimizes sensitivity over a wider frequency range.

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John,

That's my blog and antenna design. Fractal designs are used to fit a larger antenna in a smaller space. The self similar pattern has been found to reduce loss of gain when needing to compress the size of an antenna.

The wikipedia page on Fractal Antennas might be a useful read for you. http://en.wikipedia.org/wiki/Fractal_antenna

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