| The Mandelbrot set is a subset of the complex plane.
Take the sequence of complex numbers given by
2
a[0] = x a[n+1] = a[n] + y n = 0,1,2,3,...
Call this sequence a(x,y).
Given such a sequence, the set of points { a[n] | n = 0,1,2,...}
is either a bounded set or an unbounded set.
"The" Mandelbrot set is the set of all those complex
numbers z such that the sequence a(0,z) is bounded.
[It could just as well have been the sequence a(z,z)
which is the sequence a(0,z) minus its first element.]
It can be shown that with the exception of the sequence
a(0,-2) = 0, -2, -2, -2, -2, ... that the set of points
is unbounded if and only if it contains a complex number
z with |z| >= 2.
I am less sure about this next part; but I think the
set of z such that the sequence a(c,z) is bounded for
any other complex c is also called "a" Mandelbrot set;
and the set of complex z such that a(z,c) for a fixed
complex number c is bounded is called a Julia set.
There is a notes conference on the Mandelbrot set (and
programs that draw images of it) at TLE::MANDELBROT.
Dan
|
| re .2
>> There is a notes conference on the Mandelbrot set (and
>> programs that draw images of it) at TLE::MANDELBROT.
A while back this moved, along with all of the other notes
conferences on node TLE, to node TURRIS. So make that
Notes> ADD ENTRY TURRIS::MANDELBROT
or use KP7 or select to add.
Dan
|
| re .2
>> It can be shown that with the exception of the sequence
>> a(0,-2) = 0, -2, -2, -2, -2, ... that the set of points
>> is unbounded if and only if it contains a complex number
>> z with |z| >= 2.
Oops, make that, it can be shown that with the exception
of the sequence a(0,-2) = 0, -2, 2, 2, 2, ... that the
set of points is unbounded if and only if it contains a
complex number z with |z| >= 2.
>> I am less sure about this next part; but I think the
>> set of z such that the sequence a(c,z) is bounded for
>> any other complex c is also called "a" Mandelbrot set;
>> and the set of complex z such that a(z,c) for a fixed
>> complex number c is bounded is called a Julia set.
The actual definitions are a little broader than that.
The second one I mentioned gives a filled in Julia set.
The actual Julia set would be the boundary in the complex
plane of the filled in Julia set.
Dan
|