Fellow Fractaliers-
The "puzzling e-mail" which Dr Ik claims he got "2
or 3 yrs ago" from "someone" was actually a series of simple questions asking
him - apropos of some of Jim's FOTD's based on "Ikenaga Functions"
- to please provide some documentation and insight (PNL no
doubt retains the same archived messages I do).
At the time, Dr. Ik denied any authorship,
relationship to, or even knowledge of (!!) the functions
to which his name was graciously applied by Muth et al.
The attached contains more info from Dr.
Ikenaga than has ever generated re the Ik Fxns, or Sets or whatever he now
wishes to call them. Glad to see his admission of parenthood, at
last.
But please, Herr Doktor, spare me (and any other
Readers with a memory) the insipid assertion that you received "a" [implying
'single'] "e-mail" which was "puzzling" [huh?????].
In fact what you received was a series of
well-documented , courteous, inquiries over almost a month, to which
you never bothered to respond to respond appropriately. It
appears you were simply too self-obsessed to do your homework.
Attempts at revisionist history are
frequently offensive. Glad to note that Dr. I's memory has been
restored.
Happy Memorial Day.
D. Freed PhD
Dallas
I contacted almost 2 yrs ago at Millersville
at whuic point he denied any recollection of asny publications other than a
single, un-downloadable Mathematica(r) Notebook which appeared to e Class
Notes.
Nice to see that whatever it is has finally spurred
Dr Ik to give us a little background on the work which has borne his name in
fractal circles for >5yrs.
----- Original Message -----
Sent: Monday, May 27, 2002 2:48 PM
Subject: [Fractint] Fwd: Ikenaga
Fractals???
> 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
>
>
>
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