I discovered them when I was a sixteen-years old youngster, now almost eleven years ago, while crafting a fractal generator software I wrote on an Atari ST machine. In that time, generating fractals was only for those that really wanted it hard enough. You had to write your own software to do it. You had to wait for hours or days while the poor 8MHz machine produced the next 320x200, 16 color image. This was long before Internet was mainstream so there were no thriving online communities to easily share the formulae with.
The mere discovery of these two formulae is quite trivial - there was no mathematical theory behind it. They don't model anything from a real world that I'm aware of. They are a product of a sixteen year old kid that finally got (even if in the end he had to write it by itself) a software that didn't equal "fractal" to "Mandelbrot set" and had a chance to shove any random formula that came to mind into it and see the results. And from all random formulae I entered most produced dull shapes - except these two. The feeling was very much like discovering a piece of gold in gravel.In the coming months and then years, the Atari machine, the floppy discs containing the software, and me have (literally) survived the war in Yugoslavia, and several dozens of movings all around my new home country, Hungary. The flow of everyday events over the years quickly pushed my fractals out of my mind. All until recently, when the memories suddenly struck back as I browsed some fractal galleries on the Web. I instantly went to loft, and waded through piles of boxes until I have found my old computer and the floppies... To my first dismay, the ol' machine was disfunct. To my second, the floppies were unreadable on a PC. The software is gone. But I wasn't ready to give up. I fired up Fractint and began to search for the formulae once again... Based on the faint memories of what they have looked like, working by trial and error, and writing serveral leaves of paper full with unsuccessful reproduction experiments, one formula finally rendered the familiar butterfly-shaped set on the screen. And since the two formulae are only slight variations of each other, the other popped up soon after.
I have discovered my butterfly-shaped fractals again after eleven years. And this time, I have a powerful means of sharing them with the rest of the world and making sure they don't fall into oblivion once again.
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