> book on mathematics that demonstrates the "beauty" of it

I'd say that you're asking wrong question. Yes, there're quite a few books intended to demonstrate the very thing you're asking for: beauty of mathematics. You are even likely to find some showcases interesting, might be somewhat beautiful. But essentially it won't be the same "beauty" that mathematicians see. It will be more like a thousand more-or-less scientific examples of how number φ appears in different places in our life, which isn't the essence of it obviously.

So if you're scared by "summation" symbol you're better to reconsider or forget about understanding "beauty of mathematics" forever. Because "beauty of math" is math itself.

About FFT: are you sure that you mean FFT? Because it's merely an algorithm to compute Discrete Fourier Transform. If you want to understand the Fourier Transformation itself, here's basic explanation: http://betterexplained.com/articles/an-interactive-guide-to-...

Anyway, here're some easy-to-read books about math considered good by various people:

  Hardy, "Mathematician's Apology"
  Hofstadter, "Gödel, Escher, Bach"
  Klein, "Mathematics and the Search for Knowledge" (he wrote many sci-pop books which I expect to be good, but I read only few of them, so I'll recommend only this one)
  Сонин, "Постижение совершенства" — a good one, but I don't know if it was ever translated to english, sorry