[jason]

On Wed, 16 Jan 2013 03:07:17 -0500 "Soundhaspriority"
wrote in article


"Jason" wrote in message
...

On Tue, 15 Jan 2013 22:50:58 -0500 "Soundhaspriority"
wrote in article dOadnfniVaVJvmvNnZ2dnUVZ_u-


"Tobiah" wrote in message ...

I was looking into the Mandelbrot set as a source
for material in a piece that would use computer
algorithms to dictate many of the parameters of a piece.

Well, I didn't finish that project tonight, but I
went as far as to look up fractals on wikipedia. I found
some great ideas. I phoned my friend who is more of an
electronic engineer. My question was as to why the
explanation of the Mandelbrot set involved imaginary
numbers. I have a cursory college knowledge of how
they operate, but I have no visceral experience of
why we employ them in math. He mentioned something
about signal theory, and I mentioned 'impedance' and
he seemed to multiply my enthusiasm at that point.


Here it is RAP! Tell my why I must understand
the idea of irrational numbers if I am to understand
all that goes on under my recording desk.

I know you can do it!!

Weeellllll.....
There is something related that is a little easier to understand.
But
first, note that imaginary numbers, and irrational numbers, are different.
There can be imaginary irrationals, and real irrationals.

The origin of complex numbers, x + i*y, came about as a shorthand
for
solving pairs of trig equations. Suppose you have a pendulum that traces
out
a path in the x-y plane. The pendulum describes a circle in the x-y plane
if
the x and y oscillations are 90 degrees out. Instead of solving two
separate
equations, one for x, and the other for y, imaginary numbers can be used.

e ^ (2*pi*i*t) = cos (2*pi*t) + i*sin(2*pi*t)
describes a pendulum that goes around in a perfect circle once per unit t.
After the solution is obtained in this manner, the x and y motions can be
separated out.

In electronics, the same shorthand, solving two equations at one time, is
used. The two coupled equations a
a. the oscillating part, that conserves energy. This is the real part of
the
solution
b. the dissipating part, that eventually sucks all the energy out of the
circuit. This is the imaginary part.

The circle described above may be the simplest example of a Mandlebrot
set.
It orbits around the origin, 0, without going to infinity or zero. It is a
very boring circle of a Mandelbrot set. Other examples are more
interesting.
Using the same kind of notation, the path is not a simple circle, but
something which changes with every turn, eventually hitting infinity or
zero, but in the meantime describing patterns that are marvelous to look
at.

You haven't specified how you want to use these. Do you want to make a
transformation from the visual to the audio domains?

Bob Morein
(310) 237-6511

That's a really nice explanation. Thanks.

I didn't understand the OP's intent either. ...yet, it's interesting to
wonder if some of the fractal math that's been applied in digital imaging
might have a parallel in the audio domain in noise reduction or more
exotic imponderables like reverb removal.

I did some graphics programming at IBM for Mandelbrot's books and
presentations and got to know him. He didn't see the world quite the same
way that you and I do. He'd be the first to ask about audio. His early
work, that led to the fractal geometry theory, had to do with noise
reduction in telephone systems.

How long is the coastline of Britain?

It depends upon the scale, but after my brief reply, it occurred to say
more. You've posed the next challenge for Tobiah, and a challenge of
communication for explainers. The example of the circle introduces complex
numbers. If Tobiah feels comfortable with that, here's the next step:

1. Pick a complex number, x+ i*y and stick it on a piece of "paper"
2. You do something with that number, an operation, to turn it into
something else. Not just any operation, but a particular one.
3. You keep doing it forever, adding each point to the paper.
4. It builds up a set of points that form the "edge".
5. The edge is infinitely rough. No matter how much you magnify the edge, it
has a jagged appearance. In fact, the degree of jaggedness is identical no
matter what scale you look at it. This is why the Mandelbrot set is a kind
of fractal.

Bob Morein
(310) 237-6511

Yup. Self-similar at all scales. (A totally OT aside: I read recently
that cosmologists have concluded that the distrubtion of galaxies in the
Universe is NOT fractal. Go figure...) This stuff fascinates me, because
it made it easy to write comp graphics programs that produced realistic
looking terrain, plants, etc etc.