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From: scott@bu-cs.UUCP
Newsgroups: comp.graphics
Subject: Re: What's a Julia set?
Message-ID: <7660@bu-cs.BU.EDU>
Date: Fri, 15-May-87 10:29:59 EDT
Article-I.D.: bu-cs.7660
Posted: Fri May 15 10:29:59 1987
Date-Received: Sat, 16-May-87 17:30:31 EDT
References: <165500002@uiucdcsb>
Organization: Boston U. Comp. Sci.
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In-reply-to: robison@uiucdcsb.cs.uiuc.edu's message of 13 May 87 21:35:00 GMT
Posting-Front-End: GNU Emacs 18.41.4 of Mon Mar 23 1987 on bu-cs (berkeley-unix)


In <165500002@uiucdcsb>, robison@uiucdcsb.cs.uiuc.edu writes
>In the May 1987 issue of IEEE Computer Graphics, p.7, Figure 4 has the caption:
>
>        "Julia sets for Newton's method applied to polynomials of
>degree 3
>         (small spheres) and degree 4 (large spheres)."
>
>What is a "Julia set"?  Can someone point out a reference?

In this context, a Julia set is the boundary between the basins of
attraction of the roots (and any other attracting periodic orbits).
That is, all the points that do not eventually get sucked in to a root
of the polynomial (or an attr. periodic orbit) are in the Julia set.
[Notice that I keep throwing in attracting periodic orbits -- These
are what cause Newton's method to fail for an open set of initial
points]

More generally, one definition of the Julia Set is "the closure of the
repelling periodic points".  A nice reference with lots of pictures is
_The_Beauty_of_Fractals_ by H.O. Peitgen & P. Richter (Springer-Verlag).

This whole business is part of the field of Complex Dynamics, in which
there's a lot of mathematics going on lately. I haven't brought up any
more mathematical references, (though there's lots), since you're
obviously not familiar with the field.  A name generally known in
graphics circles is Mandlebrot (and the Mandlebrot set), which is
intimitely related to Julia Sets. 

Hope that helped some.