>   Don't confuse the 4-dimensional julia fractals (which the julia_fractal
> object is) with regular 2-dimensional julia fractals.
>   If you want regular 2-dimensional julia fractals, there's a pattern for
> that in MegaPov.

I don't exactly know the difference, but I guess a 4D fractal is a repetition of
a fractal design in space, thus making a 3D object... am I right?

For the 2D fractals, I would prefer to construct them myself...

Thanks,
	Simon

-- 
+-------------------------+----------------------------------+
| Simon Lemieux           | Website : http://www.666Mhz.net  |
| Email : Sin### [at] 666Mhznet | POV-Ray, OpenGL, C++ and more... |
+-------------------------+----------------------------------+

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Simon Lemieux wrote:
> 
> I don't exactly know the difference, but I guess a 4D fractal is a repetition of
> a fractal design in space, thus making a 3D object... am I right?
> 

The 4D Julia set is a true 4D object, i.e. it has one more dimension besides
height, width and depth. Obviously POV couldn't render this (and most people
can't even visualise it), so instead it renders a 3D "slice" of this 4D object -
much like a circle is a 2D slice of the sphere (3D).

-- 
Margus Ramst

Personal e-mail: mar### [at] peakeduee
TAG (Team Assistance Group) e-mail: mar### [at] tagpovrayorg

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Simon Lemieux <lem### [at] yahoocom> wrote:
: I don't exactly know the difference, but I guess a 4D fractal is a repetition of
: a fractal design in space, thus making a 3D object... am I right?

  Nope.

  A "regular" 2-dimensional julia uses complex numbers. If you are interested,
the formula is the following:

  For each point c in the complex plane, the julia set (let's call it J) will
be:

  J = { c |  lim   Z(n) != inf }
            n->inf

where:

  Z(0) = c
  Z(n) = Z(n-1)^2 + Y

where Y is a complex number (in a Mandelbrot set it will be c itself, in a
Julia set it's a chosen complex number which doesn't change).

  When the Julia set is displayed in the complex plane (it's usually
denoted so that the x-axis in a cartesian system denotes the real part of
the number and the y-axis denotes the imaginary part) it forms the peculiar
shape you know.
  (The colors displayed outside the set are not part of the set but are
created by a simple trick.)

  Now, the 4-dimensional julia uses either hypercomplex or quaternion
numbers instead of complex numbers.
  The formula is exactly the same as above, the only difference being that
hypercomplex and quaternion numbers have 4 parts instead of 2.
  This means that the set will be formed in the 4-dimensional space.
  The difference between hypercomplex and quaternion numbers is that for
numbers with more than 2 parts some mathematical operations (such as
multiplication) are not unambiguously defined. Hypercomplex and quaternion
numbers use different type of multiplication.

  Since the set has 4 dimensions, a 3-dimensional "slice" has to be taken
from the set in order to represent it in 3D space.
  This is similar to cutting a 3D object with a plane and getting a 2D shape
in the plane. But instead of making a 2D slice from a 3D object, we are
making a 3D slice from a 4D object.
  It's not possible to represent a 4D object in itself because our brain
can't handle that information. This is why we need a 3D slice of that object.

: For the 2D fractals, I would prefer to construct them myself...

  Why, when there's already a pattern for that?

-- 
main(i,_){for(_?--i,main(i+2,"FhhQHFIJD|FQTITFN]zRFHhhTBFHhhTBFysdB"[i]
):_;i&&_>1;printf("%s",_-70?_&1?"[]":" ":(_=0,"\n")),_/=2);} /*- Warp -*/

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"Warp" <war### [at] tagpovrayorg> wrote in message
news:39f3fb50@news.povray.org...

> akhgashtahsg  asdhwajhe mxzhfweaht sadfhaw hasjd

my brain is bleeding out of my ears - is this normal?

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>   A "regular" 2-dimensional julia uses complex numbers. If you are interested,
> the formula is the following:
> 
>   For each point c in the complex plane, the julia set (let's call it J) will
> be:
> 
>   J = { c |  lim   Z(n) != inf }
>             n->inf
> 
> where:
> 
>   Z(0) = c
>   Z(n) = Z(n-1)^2 + Y
> 
> where Y is a complex number (in a Mandelbrot set it will be c itself, in a
> Julia set it's a chosen complex number which doesn't change).
> 
>   When the Julia set is displayed in the complex plane (it's usually
> denoted so that the x-axis in a cartesian system denotes the real part of
> the number and the y-axis denotes the imaginary part) it forms the peculiar
> shape you know.
>   (The colors displayed outside the set are not part of the set but are
> created by a simple trick.)
> 
>   Now, the 4-dimensional julia uses either hypercomplex or quaternion
> numbers instead of complex numbers.
>   The formula is exactly the same as above, the only difference being that
> hypercomplex and quaternion numbers have 4 parts instead of 2.
>   This means that the set will be formed in the 4-dimensional space.
>   The difference between hypercomplex and quaternion numbers is that for
> numbers with more than 2 parts some mathematical operations (such as
> multiplication) are not unambiguously defined. Hypercomplex and quaternion
> numbers use different type of multiplication.
> 
>   Since the set has 4 dimensions, a 3-dimensional "slice" has to be taken
> from the set in order to represent it in 3D space.
>   This is similar to cutting a 3D object with a plane and getting a 2D shape
> in the plane. But instead of making a 2D slice from a 3D object, we are
> making a 3D slice from a 4D object.
>   It's not possible to represent a 4D object in itself because our brain
> can't handle that information. This is why we need a 3D slice of that object.

Cool, Thanks a lot!

> : For the 2D fractals, I would prefer to construct them myself...
> 
>   Why, when there's already a pattern for that?

Because, I could play with colors, and paterns and shape, make it move, etc... 
I would do this using OpenGL, so it would be fairly fast, fullscreen and quite
amazing!

Thanks,
	Simon
-- 
+-------------------------+----------------------------------+
| Simon Lemieux           | Website : http://www.666Mhz.net  |
| Email : Sin### [at] 666Mhznet | POV-Ray, OpenGL, C++ and more... |
+-------------------------+----------------------------------+

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Excellent description.  Thank you.

One common trick is to vary the value of the fixed parameter for animation.

This can also be thought of as using time to represent the fourth dimension.

Some interesting results depending on which value (and range) you choose.

Look forward to seeing some fractals in p.b.a and p.b.i soon :-)

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Tom Melly wrote:

> > nxutnfugnuft  nfqujnwur zkmusjrnug fnqsunj unfwq

> zl oenva vf oyrrqvat bhg bs zl rnef - vf guvf abezny?

Yes.

-- 
Ken Tyler

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Wasn't it Simon Lemieux who wrote:
>>   Don't confuse the 4-dimensional julia fractals (which the julia_fractal
>> object is) with regular 2-dimensional julia fractals.
>>   If you want regular 2-dimensional julia fractals, there's a pattern for
>> that in MegaPov.
>
>I don't exactly know the difference, but I guess a 4D fractal is a repetition of
>a fractal design in space, thus making a 3D object... am I right?

No. 2D Julia fractals are created by performing certain mathematical
operations using complex numbers, and 4D Julia fractals are created by
performing the same sort of mathematical operations on quaternions.

Complex numbers have two dimensions - one real and one imaginary - so
any fractals generated with them are 2D.

Quaternions have four dimensions - one real and three imaginary - so any
fractals generated with them are 4D.

When you render a 4D fractal, what you see on the screen is a 3D "slice"
through the 4D object.

-- 
Mike Williams
Gentleman of Leisure

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Simon Lemieux <lem### [at] yahoocom> wrote:
: Because, I could play with colors, and paterns and shape, make it move, etc... 
: I would do this using OpenGL, so it would be fairly fast, fullscreen and quite
: amazing!

  Look at the documentation of the fractal patterns in megapov docs and see
what they have to offer to you.

-- 
main(i,_){for(_?--i,main(i+2,"FhhQHFIJD|FQTITFN]zRFHhhTBFHhhTBFysdB"[i]
):_;i&&_>1;printf("%s",_-70?_&1?"[]":" ":(_=0,"\n")),_/=2);} /*- Warp -*/

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Mike Williams <mik### [at] nospamplease> wrote:
: 4D Julia fractals are created by
: performing the same sort of mathematical operations on quaternions.

  Or hypercomplex numbers.

-- 
main(i,_){for(_?--i,main(i+2,"FhhQHFIJD|FQTITFN]zRFHhhTBFHhhTBFysdB"[i]
):_;i&&_>1;printf("%s",_-70?_&1?"[]":" ":(_=0,"\n")),_/=2);} /*- Warp -*/

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