The Fourier Toy mentioned in the article: http://toxicdump.org/stuff/FourierToy.swf
Notice how you can click and drag to change the size and initial phase of the circle widgets. Try changing the initial phase of any of the larger components without changing the size and see what happens. It goes haywire!
The first FT toy that I wrote years ago also ignored phase. It took me forever to figure out why my reconstructed images looked like crap.
Turns out you can't just throw away half of your transform data (you get frequency and phase for each component you care about) without being fabulously clever.