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Aug 15, 1992, 7:11:52 AM8/15/92

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Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

(And I don't mean the graphic presentations using heights

instead of colours - rather globes instead of circles, and

an "apple" instead of the main cardioid.)

(And I don't mean the graphic presentations using heights

instead of colours - rather globes instead of circles, and

an "apple" instead of the main cardioid.)

Are there fractal sets in C^2? (Would be 4d, wouldn't they?)

--tommy

*************************************************************

* semper idem! * "O Frabjous Day! *

* ca...@etek.chalmers.se * Callooh! Callay!" -L.C.*

*************************************************************

Aug 15, 1992, 11:38:54 AM8/15/92

to

In a previous article, ca...@etek.chalmers.se (Tommy Vaske) says:

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

>

>Are there fractal sets in C^2? (Would be 4d, wouldn't they?)

Consider the 'vanilla' Julia and Mandelbrot sets:

Z(n+1)=Z(n)**2+C

The difference is that in one, Z changes and in the other C.

Now, what if *both* Z and C varied? This makes four dimensions

(Real Z, real C, imag Z, imag C).

Fractint has a fractal called 'julibrot' that displays a 3-d slice of

this 4-d fractal.

It looks quite neat. (Rather an eye-bender, though.)

--

|Jeff Epler Additions Welcome c(-8 ;-) >{8-) |

| :) (=( =-] (-= Celebrating the variety of faces =-> :^) {-= |-) (: |

| Lincoln, Nebraska|

Aug 15, 1992, 6:41:57 PM8/15/92

to

ca...@etek.chalmers.se (Tommy Vaske) writes:

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

I saw this mentioned in the latest "Algorithm" journal...The method used was

to vary the exponent (Z <- Z^p + C) and use p as the third dimension. I didn't

get a chance to carefully read the article, but I seem to recall it saying that

something happened near the integers...Now, I am nowhere near an expert in the

fractal area, but it occured to me that Z^p would not be really well-defined

for non-integral p. The article used exp(p*ln(Z)), but in the complex numbers

ln(Z) could be one of an infinite number of things (separated by multiples of

2\pi), and if p is not an integer, you'll get different values of exp(p*ln(Z))

depending on which you choose. But like I said, I didn't read the article that

closely, so this issue might have been brought up in the article itself...If so

I apologize...

--

-- Erick, the perfect square :-)

Aug 16, 1992, 9:00:55 AM8/16/92

to

ca...@etek.chalmers.se (Tommy Vaske) writes:

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

>

>Are there fractal sets in C^2? (Would be 4d, wouldn't they?)

>

>--tommy

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

>

>Are there fractal sets in C^2? (Would be 4d, wouldn't they?)

>

>--tommy

There are fractals in the 4d realm of complex numbers, called the quaternions.

Basically, a quaternion Q = x + i*y + j*z + k*t, where i*i = j*j = k*k = -1,

and i*j = k = -j*i, j*k = i = -k*j, and k*i = j = - i*k. 3d slices of the 4d

Mandelbrot set (let the k part of c = 0) do reveal spheres instead of circular

disks. Kind of a disappointment, really, to get these extra dimensions and

not see something new and exciting (not that I've seen all of 4d fractal

space! :-)

Kerry Mitchell

Aug 17, 1992, 1:21:36 PM8/17/92

to

In article <1992Aug15.1...@etek.chalmers.se>

ca...@etek.chalmers.se (Tommy Vaske) writes:

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

ca...@etek.chalmers.se (Tommy Vaske) writes:

>Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

>(And I don't mean the graphic presentations using heights

>instead of colours - rather globes instead of circles, and

>an "apple" instead of the main cardioid.)

One can specify an Iterated Function System in any number of

dimensions, e.g. in 3d. An example is the famous fern of

Michael Barnsley, extented to 3d as shown on the inside of the

cover of his book `Fractals everywhere'.

Of course are fractals generated with IFS of linear type

(when using affine transformations, as mostly done), not

like the Mandelbrot-set that is non-linear.

--

Michael A. Neuhauser Dept. for Pattern Recoginition and Image Processing

m...@prip.tuwien.ac.at Technical Univerity of Vienna, Treitlstr. 3/183/2

A-1040 Vienna AUSTRIA

Aug 17, 1992, 2:41:07 PM8/17/92

to

ca...@etek.chalmers.se (Tommy Vaske) writes:

Does anybody know of a 3d fractal set, e.g. a 3d mandelbrot?

=============================================================

There are many ways to generate 3d fractal sets. Several other

contributors to this group have pointed out a few. Another

method which incorporates the Mandelbrot set construction is

to treat the usual Z -> Z^2 + C map of the complex plane as if

it were a map of the real plane into itself. That is, the eqns

would look like :

X -> X^2 - Y^2 + A

Y -> 2XY + B

Where Z = X + Yi and C = A + Bi .

All that's necessary to extend to 3 dimensions is to add the eqn

for the real Z component. In general, iterates of R^3 into R^3

can be written as :

X -> F(X, Y, Z)

Y -> G(X, Y, Z)

Z -> H(X, Y, Z)

Make something up !

Another kinda cool way to get 3-D fractals from iterates of maps

is to apply some simple numerical integration scheme to a 3-D flow.

For example, take the Lorenz eqns :

Xdot = -sX + sY

Ydot = -XZ + rX - Y

Zdot = XY - bZ

Where s, r, b are positive constants and dot means differentiation wrt time.

Applying the Euler forward differencing scheme of numerically integrating

these eqns with a time increment t will give a 3-D iterated map which

approximates the Lorenz eqns and gives rise to a chaotic attractor in 3-D.

-rr-

--

Verum est ... quod superius est sicut | The truth is that what is above is like

quod inferius et quod inferius est | what is below and what is below is

sicut quod superius, ad perpetrando | like what is above, to accomplish

miracula rei unius. | the miracles of the one thing.

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