question

[Sonda]

what would happend to mandelbrot set if we would change axes of surface
from (-1)^0, (-1)^(1/2) (real numbers and i ) to (-1)^(1/3),
(-1)^(1/7).

possibly we can have "true" three dimensional fractal with axes for
example
(-1)^(1/7), (-1)^(1/4), (-1)^(1/13) and sequence Zn+1=Zn^3 - Zn^2 + C.

[E***@spamblock.panix.com]

I hope that somebody answers this. I am unfamiliar with the math.

But I have wondered about 3 dimensional Mandelbrot sets.

And in principle, is it possible to calculate the set with n dimensions?
And then to project it into a 2 dimensional representation of a 3
dimensional space, like the hypercube representations one can find on the
'web?

Post by Sonda
what would happend to mandelbrot set if we would change axes of surface
from (-1)^0, (-1)^(1/2) (real numbers and i ) to (-1)^(1/3),
(-1)^(1/7).
possibly we can have "true" three dimensional fractal with axes for
example
(-1)^(1/7), (-1)^(1/4), (-1)^(1/13) and sequence Zn+1=Zn^3 - Zn^2 + C.

[Sonda]

in this (my) idea with axes (-1)^p it is possible to create
n-dimensional fractal,
but i'm asking to check with the group if my theory is correct.

Post by E***@spamblock.panix.com
I hope that somebody answers this. I am unfamiliar with the math.
But I have wondered about 3 dimensional Mandelbrot sets.
And in principle, is it possible to calculate the set with n dimensions?
And then to project it into a 2 dimensional representation of a 3
dimensional space, like the hypercube representations one can find on the
'web?

Post by Sonda
what would happend to mandelbrot set if we would change axes of surface
from (-1)^0, (-1)^(1/2) (real numbers and i ) to (-1)^(1/3),
(-1)^(1/7).
possibly we can have "true" three dimensional fractal with axes for
example
(-1)^(1/7), (-1)^(1/4), (-1)^(1/13) and sequence Zn+1=Zn^3 - Zn^2 + C.

--
A nation of sheep will beget a government of wolves.
--Edward R. Murrow

[Sonda]

complex number a+b*i==a*(-1)^0+b*(-1)^(1/2)

[Roger Bagula]

z'=Exp[I*2*Pi/m]*(z^2+c)
Rotaion by a cyclotomic angle power of 1/m: 360 degrees/ m
z[n]=Exp[I*2*Pi/m]*(z[n-1]^3-z[n-1]^2+c)

You have to distinguish what you mean by "3d dimensional".
Regular Mandlbrots are 3d if you take the iteration color as height.
I doubt this is your meaning.

There is a program you can buy for doing quaternion Mandelbrots and
Julias by Terry Gintz.
Quaternions are sometimes called "hypercomplex numbers"
Akira bergman has also his H-numbers.
It is called Clifford algebra after the guy who came after Hamilton
and increased the number of dimensions.
There are octonions with eight dimensions even.
http://www.google.com/search?hl=en&lr=&safe=off&q=octonions+fractals&btnG=Search
Basically:
z1=x1+I*y1
z2=x2+I*y2
q=x1+I*y1+J*(x2+I*x2)=x1+I*y1+J*x2+K*x2
J*I==K
I^2=-1
J^2=-1
k^2=-1
q*qstar=(x1+I*y1+J*x2+K*x2)*(x1-I*y1-J*x2-K*x2)=x1^2+y1^2+x2^2+y2^2
Lord Hamilton's quaternions were first created to make spherical
geometry easier...
They turn out what one might call taffy pull sets in 3d.
http://www.google.com/search?hl=en&q=terry+Gintz+quaternions&btnG=Google+Search
http://home.att.net/~d.a.hasson/quasz_for_dummies.html
In four dimensions you have formulas like:
q[n]=Exp[I*2*Pi*i]/L]*Exp[J*2*Pi*i]/M]*Exp[K*2*Pi*i]/N]*(q[n-1]^2+c)
That's three orthogonal directions of rotation instead of just one.

Post by Sonda
what would happend to mandelbrot set if we would change axes of surface
from (-1)^0, (-1)^(1/2) (real numbers and i ) to (-1)^(1/3),
(-1)^(1/7).
possibly we can have "true" three dimensional fractal with axes for
example
(-1)^(1/7), (-1)^(1/4), (-1)^(1/13) and sequence Zn+1=Zn^3 - Zn^2 + C.

[Sonda]

Post by Roger Bagula
z'=Exp[I*2*Pi/m]*(z^2+c)
Rotaion by a cyclotomic angle power of 1/m: 360 degrees/ m
z[n]=Exp[I*2*Pi/m]*(z[n-1]^3-z[n-1]^2+c)
You have to distinguish what you mean by "3d dimensional".
Regular Mandlbrots are 3d if you take the iteration color as height.
I doubt this is your meaning.

by 3 dimensional fractal i mean fractal that exists in space with three
axis
x, y, z.

axis x is in form "real number" * 1
axis y is in form "real number" * (-1)^(1/2) // where (-1)^(1/2) == i
== square of (-1)
axis z is in form "real number" * (-1)^(1/3)

it can be generalization of complex number

Post by Roger Bagula
There is a program you can buy for doing quaternion Mandelbrots and
Julias by Terry Gintz.
Quaternions are sometimes called "hypercomplex numbers"
Akira bergman has also his H-numbers.
It is called Clifford algebra after the guy who came after Hamilton
and increased the number of dimensions.
There are octonions with eight dimensions even.
http://www.google.com/search?hl=en&lr=&safe=off&q=octonions+fractals&btnG=Search
z1=x1+I*y1
z2=x2+I*y2
q=x1+I*y1+J*(x2+I*x2)=x1+I*y1+J*x2+K*x2
J*I==K
I^2=-1
J^2=-1
k^2=-1
q*qstar=(x1+I*y1+J*x2+K*x2)*(x1-I*y1-J*x2-K*x2)=x1^2+y1^2+x2^2+y2^2
Lord Hamilton's quaternions were first created to make spherical
geometry easier...
They turn out what one might call taffy pull sets in 3d.
http://www.google.com/search?hl=en&q=terry+Gintz+quaternions&btnG=Google+Search
http://home.att.net/~d.a.hasson/quasz_for_dummies.html
q[n]=Exp[I*2*Pi*i]/L]*Exp[J*2*Pi*i]/M]*Exp[K*2*Pi*i]/N]*(q[n-1]^2+c)
That's three orthogonal directions of rotation instead of just one.

Post by Sonda
what would happend to mandelbrot set if we would change axes of surface
from (-1)^0, (-1)^(1/2) (real numbers and i ) to (-1)^(1/3),
(-1)^(1/7).
possibly we can have "true" three dimensional fractal with axes for
example
(-1)^(1/7), (-1)^(1/4), (-1)^(1/13) and sequence Zn+1=Zn^3 - Zn^2 + C.

[Maarten van Reeuwijk]

Post by Sonda

Post by Roger Bagula
z'=Exp[I*2*Pi/m]*(z^2+c)
Rotaion by a cyclotomic angle power of 1/m: 360 degrees/ m
z[n]=Exp[I*2*Pi/m]*(z[n-1]^3-z[n-1]^2+c)
You have to distinguish what you mean by "3d dimensional".
Regular Mandlbrots are 3d if you take the iteration color as height.
I doubt this is your meaning.

by 3 dimensional fractal i mean fractal that exists in space with three
axis
x, y, z.
axis x is in form "real number" * 1
axis y is in form "real number" * (-1)^(1/2) // where (-1)^(1/2) == i
== square of (-1)
axis z is in form "real number" * (-1)^(1/3)
it can be generalization of complex number

This is not a generalization as defining i^2 = -1 is enough to deal with
cases like (-1)^(1/3), which is i^(2/3). Of course you can always choose to
add an extra dimension such that it is the imaginary axis to the power 2/3,
but it is *dependent*. The quaternions Roger spoke about are real
generalizations of complex numbers.

HTH, Maarten

[Roger Bagula]

Post by Sonda

Post by Roger Bagula
z'=Exp[I*2*Pi/m]*(z^2+c)
Rotaion by a cyclotomic angle power of 1/m: 360 degrees/ m
z[n]=Exp[I*2*Pi/m]*(z[n-1]^3-z[n-1]^2+c)
You have to distinguish what you mean by "3d dimensional".
Regular Mandlbrots are 3d if you take the iteration color as height.
I doubt this is your meaning.

by 3 dimensional fractal i mean fractal that exists in space with three
axis
x, y, z.
axis x is in form "real number" * 1
axis y is in form "real number" * (-1)^(1/2) // where (-1)^(1/2) == i
== square of (-1)
axis z is in form "real number" * (-1)^(1/3)
it can be generalization of complex number

Take a look at the pictures that come up in this link:
http://images.google.com/images?hl=en&q=mandelbrot%203d&btnG=Google+Search&sa=N&tab=wi

Sometimes it is hard to fathom the depepth of people's knowledge.
Everybody thinks they "is" the new star.
But the mathematics just keep trudging along.
It is possible to Bezier your z axis k values so that they approximate
real numbers: that is interpolate stopping points/ escape points between
the integer k's. I've done it.
That is "different" than a "true" 3d.
You need to use something like so(3) or su(3) over the reals
to get a true 3d in real numbers and it will be a fourth interation
escape number k or n.

[A N Niel]

Post by Sonda
axis x is in form "real number" * 1
axis y is in form "real number" * (-1)^(1/2) // where (-1)^(1/2) == i
== square of (-1)
axis z is in form "real number" * (-1)^(1/3)
it can be generalization of complex number

As they have been telling you, this doesn't make sense.
if you take these three:
1, (-1)^(1/2), (-1)^(1/3),
then any one of them can be written as a linear combination of the
other two.

[c***@lycos.com]

I also agree. I guess that good generalization of complex numbers are
quaternions and hypercomplex numbers. I believe both are not the same

Quaternions: i^2 = j^2 = k^2 = -1, i = jk = -kj, j = ki = -ik,
k = ij = -ji
Hypercomplex: i^2 = j^2 = -k^2 = -1, i =-jk = -kj, j = -ki = -ik, k
= ij = ji

Note that hypercomplex multiplication is commutative, while the
quaternion multiplication is not.

I "invented" my own 3D complex numbers years ago to create this
pictures http://rgba.scenesp.org/iq/fractals/f3d/f3d.htm, but later
discovered that I just had invented a slice of the well known
quaternions :(

Actually, apparently there was quite a lot of interest in 4D fractals
in the 90s, look to that paper http://www.evl.uic.edu/hypercomplex/ and
the corresponding link to the very nice PDF at
http://www.evl.uic.edu/hypercomplex/html/book/book.pdf

There has not been much more research in the last 15 years however.The
only difference is that nowadays you can draw 3D julia sets in
realtime.

Inigo Quilez

[Roger Bagula]

I ran across a defintion of the D2:
http://mathworld.wolfram.com/Vierergruppe.html
It is entirely symmetric in it's multiplication table
but is of the same order of group elements as these other:
SO(3),SU(2), quaternions and hypercomplex numbers...
As the Mandelbrot set is D1 like
and z^3+c is D2 like it might give a new way to approach 3d fractals too?
Remember these groups are based on polygons.
My little remarkable:
z'=I*c*z2+c
or the other
z'=c*(z^2+1)
by being cubic like would be a quadratic that behaves as D2 like?
I'll have to try that today.

Post by c***@lycos.com
I also agree. I guess that good generalization of complex numbers are
quaternions and hypercomplex numbers. I believe both are not the same
Quaternions: i^2 = j^2 = k^2 = -1, i = jk = -kj, j = ki = -ik,
k = ij = -ji
Hypercomplex: i^2 = j^2 = -k^2 = -1, i =-jk = -kj, j = -ki = -ik, k
= ij = ji
Note that hypercomplex multiplication is commutative, while the
quaternion multiplication is not.
I "invented" my own 3D complex numbers years ago to create this
pictures http://rgba.scenesp.org/iq/fractals/f3d/f3d.htm, but later
discovered that I just had invented a slice of the well known
quaternions :(
Actually, apparently there was quite a lot of interest in 4D fractals
in the 90s, look to that paper http://www.evl.uic.edu/hypercomplex/ and
the corresponding link to the very nice PDF at
http://www.evl.uic.edu/hypercomplex/html/book/book.pdf
There has not been much more research in the last 15 years however.The
only difference is that nowadays you can draw 3D julia sets in
realtime.
Inigo Quilez

[Sonda]

ok

i just thought that i find reason to lern directx3d,
but if you are saying that there will be no visual effect then i try
somthing else

and i apologise for saying generalization of complex number

Post by Roger Bagula
http://mathworld.wolfram.com/Vierergruppe.html
It is entirely symmetric in it's multiplication table
SO(3),SU(2), quaternions and hypercomplex numbers...
As the Mandelbrot set is D1 like
and z^3+c is D2 like it might give a new way to approach 3d fractals too?
Remember these groups are based on polygons.
z'=I*c*z2+c
or the other
z'=c*(z^2+1)
by being cubic like would be a quadratic that behaves as D2 like?
I'll have to try that today.

Post by c***@lycos.com
I also agree. I guess that good generalization of complex numbers are
quaternions and hypercomplex numbers. I believe both are not the same
Quaternions: i^2 = j^2 = k^2 = -1, i = jk = -kj, j = ki = -ik,
k = ij = -ji
Hypercomplex: i^2 = j^2 = -k^2 = -1, i =-jk = -kj, j = -ki = -ik, k
= ij = ji
Note that hypercomplex multiplication is commutative, while the
quaternion multiplication is not.
I "invented" my own 3D complex numbers years ago to create this
pictures http://rgba.scenesp.org/iq/fractals/f3d/f3d.htm, but later
discovered that I just had invented a slice of the well known
quaternions :(
Actually, apparently there was quite a lot of interest in 4D fractals
in the 90s, look to that paper http://www.evl.uic.edu/hypercomplex/ and
the corresponding link to the very nice PDF at
http://www.evl.uic.edu/hypercomplex/html/book/book.pdf
There has not been much more research in the last 15 years however.The
only difference is that nowadays you can draw 3D julia sets in
realtime.
Inigo Quilez

[c***@lycos.com]

what you mean? There is LOT of visual fun on rendering 3d fractals...
Directx is a good way to display them, as well as OpenGL (prefered to
me)

Inigo Quilez

[dave]

Hi Sonda,

In addition to Quaternions/Hypercomplex/Octonions etc.
you can view the complex Mandelbrot set (and many of the
Julias) together in 3D by using any of the 3 of 4 axes for z
and c in z^2+c - generally referred to as the 3D Julibrot.

I've just been updating my old 3D formulas for Ultrafractal,
you can see some results here:
http://makinmagic.deviantart.com/
and here:
http://www.renderosity.com/gallery.ez?ByArtist=Yes&Artist=MakinMagic

bye
Dave

Post by Sonda
what would happend to mandelbrot set if we would change axes of surface
from (-1)^0, (-1)^(1/2) (real numbers and i ) to (-1)^(1/3),
(-1)^(1/7).
possibly we can have "true" three dimensional fractal with axes for
example
(-1)^(1/7), (-1)^(1/4), (-1)^(1/13) and sequence Zn+1=Zn^3 - Zn^2 + C.