From: Bruce Ikenaga <bikenaga@bellatlantic.net> Subject: Ikenaga fractals??? Hi- Well, in a fit of hubris, I did a Google search on my name and turned up the page: http://home.att.net/~Paul.N.Lee/FotD/FotD_01-01-21.html which has a link to my home page, so I assume you're referring to me. I recalled getting a puzzling email from someone two or three years ago asking me about the "Ikenaga set", and I told the guy I had no idea what in the world he was talking about. But when I saw the formula on another page http://www.geocities.com/CapeCanaveral/Launchpad/5113/fr25.htm it jogged my memory. This is the stupid thing that appeared in A. K. Dewdney's Scientific American column, isn't it? Oh geez. In the first place, I never did (what mathematicians would call) research on this. Back around 1984, I heard about the Mandelbrot set --- a friend actually did some of the programming for John Hubbard, who with Adrien Douady did a lot of the *real* work on it about that time. So I was cranking out pictures of the M-set, first on the DEC Pro-350 in the math department at CWRU, and later on my Tandy 2000 (my first computer!). I printed out nifty color pictures on my old CGP-220 color inkjet (one of the first commercially available). Mathematicians generalize at the drop of a hat, so one day I asked Mike Hurley, the resident dynamical systems guy, what would be the cubic "equivalent" of the M-set. Mike told me that up to topological conjugacy (if I recall correctly), you should use the one-parameter cubic z^3 + (c - 1)z - c. Without thinking about it, I just dumped that into the Mandelbrot program in place of z^2 + c, and was soon happily cranking out lots of pictures that looked like cacti. I thought Dewdney might find it amusing, so I sent him some. I was surprised when I saw my name mentioned in his column ... ... and also embarassed. Because I showed Mike the stuff, and when he asked me what I did, he said "That's not right". See, I just put the cubic in place of the quadratic, so the program was still iterating z = 0. While you do get pictures, they aren't the "right" pictures from the dynamical point of view. The reason you iterate z = 0 with the quadratic is that it's a critical point of f(z) = z^2 + 1. By analogy, for the cubic you should iterate the critical points (plural) and (I hope I don't screw this up!) see if *at least one* escapes. (If I'm wrong, then you see if *both* escape; unfortunately, I know nothing about dynamics. As I said, I don't work in the area!) The derivative for the cubic is 3z^2 + (c - 1). So you need to set this equal to 0 and solve for z (which gives two complex square roots, which gives two critical points). You iterate these guys, not z = 0. Thus, you'll be iterating different points, depending on the current value of c. So I fixed my program and got new pictures which don't look like cacti --- not surprisingly, they look more like the quadratic M-set --- and quickly wrote Dewdney and told him I screwed up. But he never got a chance to print a correction, though I don't believe the original article actually said in detail how the set was generated. (If it had, I'm sure any number of mathematicians would have written in to point out that it was the wrong set.) I guess people read the column and just assumed you did the same thing as with the quadratic. I got bored with making pictures of fractals after two or three years. (A lot of other mathematicians got bored as well. As an amusing example, the October 1992 issue of the American Mathematical Monthly has a picture of the Mandelbrot set on the front cover. Inside, it says: "Cover: The Mandelbrot set. Neither it nor the word fractal is mentioned anywhere in this issue of the Monthly, except on the inside front cover." (Kind of like 10 years of the "X-Files", I guess.) (As an aside, the last pictures I did were from a paper by Grebogi, Ott, and Yorke. Not fractals, but some of the best black-and-white wallpaper ever! I've never seen those pictures since, so maybe when I have time I'll dig up the program and put it up on my page so people can play.) Anyway, I'm chagrined to find something apparently named after me, when as a mathematician I know it's the "wrong" object! I would greatly appreciate it if you let people know ... though I guess most are just happy that the thing makes pretty pictures, huh? :-( Feel free to copy this to others if it helps to correct this. Oh ... do any of those fractal-generating programs actually do the cubic, but the *correct* way? If not, you could do a favor for the people who write these things by letting them know. (Maybe go ask a dynamical systems person first whether you want *at least one* or whether you want *both* critical points to escape in order to color the point c. Don't take my word for it --- as you can see, I'm not a good source for this!) I hope this somewhat lengthy explanation helps. Bruce Ikenaga Department of Mathematics Millersville University Millersville, PA 17551