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Oct 1, 1986, 8:55:38 PM10/1/86

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Has anyone tried to generalize the Mandelbrot set to include a third dimension?

All the examples I've seen have been limited to the complex plane. Is there

any such thing as a Mandelbrot solid?

All the examples I've seen have been limited to the complex plane. Is there

any such thing as a Mandelbrot solid?

Oct 2, 1986, 12:18:23 PM10/2/86

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In some ways, yes. The Julia sets are related to the Madlebrot set, and they

can be used for two kinds of generalizations to higher dimensions. First, some

definitions:

If c is a complex number and f_c(z) = z^2 - c, the Julia set associated with c

is the set of all z such that the sequence z,f_c(z),f_c(f_c(z)),... does not

converge to infinity. The Mandlebrot set is the set of all c for which the

Julia set obtained from c has non-zero area (or it's the closure of that set; I

can't remember which). A powerful theorem by Douady and Hubbard says that the

Mandlebrot set is also the set of c for which 0 is in the Julia set for c.

So, in four dimensions, you can draw the set of all c and z for which

z,f_c(z),f_c(f_c(z)),... does not converge to infinity, and in addition to

getting all of the Julia sets, the Mandlebrot set will appear as a

two-dimensional cross-section. Mandlebrot's book has some three-dimensional

cross-sections of this set.

The other way to generalize is quaternions. For a given quaternion c, you can

look at look at the set of all quaternions z for which z,f_c(z),f_c(f_c(z)),...

does not escape to infinity; one two-dimensional cross-section will be the

conventional Julia set. Mandlebrot's book also has 3-D cross sections of these

sets.

Unfortunately, the set of quaternions c for which 0,f_c(0),f_c(f_c(0)),...

does not escape to infinity is just many copies of the conventional Mandlebrot

set. The reason is that a quaternion, when multiplied by a real number or a

power of itself, behaves just like a complex number. I don't know if Douady

and Hubbard's theorem generalizes to quaternions.

I must add that although Mandlebrot's book, "The Fractal Geometry of Nature",

has some nice pictures, most of it is mathematically inane. Papers by Douady

and Hubbard are probably more interesting, albeit more difficult, reading.

gregregreg

Oct 3, 1986, 11:22:20 PM10/3/86

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In article <3...@oblio.UUCP> p...@oblio.UUCP (Paul Fronberg) writes:

>Has anyone tried to generalize the Mandelbrot set to include a third dimension?

Yes, look for references to Alan Norton. He explained them to me over

greasy hamburgers in a cheap fast food restaraunt once. The Smithsonian

Magazine had one of the little root-like solids on their cover a long time

ago...

Douglas

ARPA: tra...@locus.ucla.edu

UUCP: ..!{sch-loki,silogic,randvax,ihnp4,sdcrdcf,trwspp,ucbvax}!ucla-cs!trainor

SNAKE: tra...@eric.sidewinder.snake

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