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From: greg@endor.harvard.edu (Greg)
Newsgroups: net.graphics,net.math
Subject: Re: 3D Mandelbrot sets?
Message-ID: <301@husc6.HARVARD.EDU>
Date: Thu, 2-Oct-86 12:18:23 EDT
Article-I.D.: husc6.301
Posted: Thu Oct  2 12:18:23 1986
Date-Received: Sat, 4-Oct-86 05:47:55 EDT
References: <324@oblio.UUCP>
Sender: news@husc6.HARVARD.EDU
Reply-To: greg@endor.UUCP (Greg)
Organization: Harvard
Lines: 38
Keywords: Mandelbrot, Fractals
Xref: watmath net.graphics:1956 net.math:3665

In article <324@oblio.UUCP> paf@oblio.UUCP (Paul Fronberg) writes:
>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?

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