That is how the Mandelbrot set came to be. We drew this set as a - well, we call it a map; on this map we should see in which parts of a plane different behaviours occurred. The first impression of this map, of course, was extraordinary complication. The first impressions we had were of shapes that were dirty. Now, I must emphasise, what kind of instruments I was using. This work was started when I was at IBM, and continued when I was a visiting professor of mathematics at Harvard. Harvard then had very poor computer facilities, extremely poor ones. We had a small DEK computer, and we had a Tektronix screen, a graphic screen, which was very old and very worn out, and a Versatec printer, with a very old-fashioned processor which gave rise to very dirty pictures under ordinary conditions. So the first drawing of what became known as the Mandelbrot set a few years later, was very, very messy, and I thought it was the fault of our terrible equipment. Well, to make sure, we zoomed; that is, we looked in greater detail at a small part of the field. Some pieces were clearly dirt; other pieces we didn't know. I took advantage of a trip back to IBM and went to beg my previous assistant, and remind him that I was a nice person to work for, that I let him do all kinds of things which I didn't care about because that would help his career after he stopped working for me, and finally convinced him to give me a day. So, he had some difficulty reproducing the program done at Harvard, and then he reproduced it: their machines were far better, much faster; the screen was extremely high quality; and I was expecting a nice clean picture. The picture was worse than ever. It was a very peculiar cardioid, then a circle, then this and that, and then at the boundary- mess beyond imagination. Well I couldn't beg Mark Laff to continue, so I went back to Harvard, then started looking at the boundaries in greater detail and greater detail. At some point - we had picture after picture; we repeated the picture - and the dirt was different. Where it was different in different transparencies, probably it was dirt. And we zoomed in, - it was dirt. In other places, a strange thing was happening. We saw the same piece of dirt on top and it was symmetrical at the bottom. That was very unlikely to happen because of just a bad system. We looked at these pieces in greater detail: total shock. Those pieces were exactly like the whole thing but smaller and a little bit deformed. Well, we looked at some other pieces in greater detail and then we started taking values of the parameter c, in z2+c. Actually, I was using a different formalism, but it was equivalent. When c is in those pieces - I called them islands - and when it was there, inside, we got a limit cycle; when it was outside we didn't get a limit cycle. More precisely, the thing was going on moving around and we couldn't see whether it was a very big cycle or it was no cycle: therefore, chaos.
