Does anyone know why POV-Ray documentation states the following?:

"Of the two, the quaternions are much better known, but one can argue that
hypercomplex numbers are more useful for our purposes, since complex valued
functions such as sin, cos, etc. can be generalized to work for
hypercomplex numbers in a uniform way."

Long ago, I took as an exercise to write a document about the way every
elementary function could be extended to Hamilton's quaternions, by
recalling my lessons of Linear Algebra and the fact that the set of complex

the sum of a diagonal and an antisymmetric one, and applying the same rule

study this matter, and even a recent thread about a bunch of macros to work
with quaternions provided trigonometric and exponential functions for them,
as well as their inverses. So why do quaternion fractals have to be
restricted to quadratic and cubic functions? I'm also wondering why there
seems to be no engine capable of rendering quaternion Mandelbrot sets...

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"Frango com Nata" <nomail@nomail> wrote:
> Does anyone know why POV-Ray documentation states the following?:
>
> "Of the two, the quaternions are much better known, but one can argue that
> hypercomplex numbers are more useful for our purposes, since complex valued
> functions such as sin, cos, etc. can be generalized to work for
> hypercomplex numbers in a uniform way."
>
> Long ago, I took as an exercise to write a document about the way every
> elementary function could be extended to Hamilton's quaternions, ....

Hi,

this has been discussed before, multiple times probably. I brought up the
question myself 3.5 years ago, though the thread strangely doesn't show
when you search here for "julia quaternion", see here :
http://news.povray.org/povray.general/thread/%3C3cdb6b98%40news.povray.org%3E/

The discussion then continued in personal mails between Peter Popov and me,
and the short version is:
- the original author of the quaternion stuff seems to have disappeared from
the net
- no active member (THEN, might have changed...) of the POV-team really
understands the quaternion-relevant code well enough to try to implement
new types
- there is a bug in the quaternion rendering that nobody can fix, for the
same reasons as above

>  I'm also wondering why there
> seems to be no engine capable of rendering quaternion Mandelbrot sets...

One reason might be, that the "classical" Mandelbrot sets of formulas
pow(z,n)+c should look quite boring because any 3d-cut through them
orthogonal to the real axis is just a sphere ...

Greetings

Karl

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Karl Anders wrote:

 > Hi,


Hi :)

 > this has been discussed before, multiple times probably. I brought up the
 > question myself 3.5 years ago, though the thread strangely doesn't show
 > when you search here for "julia quaternion", see here :
 > 
http://news.povray.org/povray.general/thread/%3C3cdb6b98%40news.povray.org%3E/


Well, now that you show me its address, I remember reading that thread 
shortly after it was created, when I'd been introduced to POV-Ray for 
just a few months, and couldn't access the Internet from my own computer 
to start writing here...

Since them, I've read a little about quaternions, namely a page titled 
"Doing Physics with Quaternions" 
(http://world.std.com/~sweetser/quaternions/qindex/qindex.html), where I 
learnt that approach to them as the pair composed by a real number and 
real triplet, quite suitable for representing time and space. As was to 
be expected, your definition for the quaternion exponential function is 
identical to the one I came up with by resorting to the corresponding 4 

Lagrange-Sylvester interpolation polynomial, and the same happens 
regarding circular and hyperbolic trigonometric functions, as they can 
be defined in terms of exponentials. I've even written several macros to 
handle quaternions with all those functions and their inverses, as well 
as operating with them.

And now that I remembered this issue of quaternions in POV-Ray, I 
Google-searched the whole site, but couldn't find a single thread about 
it, so felt nothing was left but asking again from scratch...

 > The discussion then continued in personal mails between Peter Popov 
and me,
 > and the short version is:
 > - the original author of the quaternion stuff seems to have 
disappeared from
 > the net
 > - no active member (THEN, might have changed...) of the POV-team really
 > understands the quaternion-relevant code well enough to try to implement
 > new types
 > - there is a bug in the quaternion rendering that nobody can fix, for the
 > same reasons as above


What a pity... But I've seen Fractal Explorer and Quat, both of which 
provide a slightly wider variety of iterating functions, but they're all 
defined by a finite number of quaternion sums and products, or at most, 

transcendental function like the exponential, and I find this at least 
curious, and a bit disappointing...

 >> I'm also wondering why there
 >> seems to be no engine capable of rendering quaternion Mandelbrot sets...


 > One reason might be, that the "classical" Mandelbrot sets of formulas
 > pow(z,n)+c should look quite boring because any 3d-cut through them
 > orthogonal to the real axis is just a sphere ...


Well, I hadn't taken the care to realize that :) But of course, it 
wouldn't be the case of the fractal defined by the same iteration 
formula q[n+1]=f(q[n])+c as Mandelbrot's, but where you could specify an 
arbitrary quaternion q[0] as the initial term, instead of just setting 
it to zero, so achieving the same single quaternion degree of freedom 
we've got when selecting a Julia fractal for some given function.

Regards.

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Frango com Nata <odo### [at] yahoocombr> wrote:
>
> Since them, I've read a little about quaternions, namely a page titled
> "Doing Physics with Quaternions"
> (http://world.std.com/~sweetser/quaternions/qindex/qindex.html), ...

interesting link, thank you. Haven't got enough time to browse it seriously
just now, too much RealLife what with christmas suddenly being so close
again, but I will try to look into it in the near future.

>
>  > One reason might be, that the "classical" Mandelbrot sets of formulas
>  > pow(z,n)+c should look quite boring because any 3d-cut through them
>  > orthogonal to the real axis is just a sphere ...
>
>
> Well, I hadn't taken the care to realize that :) But of course, it
> wouldn't be the case of the fractal defined by the same iteration
> formula q[n+1]=f(q[n])+c as Mandelbrot's, but where you could specify an
> arbitrary quaternion q[0] as the initial term, instead of just setting
> it to zero, so achieving the same single quaternion degree of freedom
> we've got when selecting a Julia fractal for some given function.
>

Well, that might give some interesting pictures, but there is mathematical
reason for "THE" Mandelbrot Set being the one with z(0) = 0; it's simply
the only one with meaning - but you probably know that, don't you ;-)

Talking about shortcomings of POV-Ray's incorporation of quaternions there
is the additional problem that you can't choose the 3d-cut you want to see
freely ( see the docs; w != 0  !!!), and that the pictures are distorted !
Last point is easily seen by depicting sqr Julia with c=0; as with complex
numbers, this defines a 4D-sphere regardless of iteration-depth, and any
"plane" 3d-cut through a 4D-sphere is a 3D-sphere ( or a point or empty ).
BUT you get ellipsoids ...

Ah well, I fear that if we say too much, we will be asked to implement that
ourselves or shut up - that has happened before to very small minorities
asking for a lot of programmer's work - and rightfully so, I fear.

Have a nice weekend

Karl

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Karl Anders wrote:
> Well, that might give some interesting pictures, but there is mathematical
> reason for "THE" Mandelbrot Set being the one with z(0) = 0; it's simply
> the only one with meaning - but you probably know that, don't you ;-)


connectivity of Julia sets? Otherwise, I don't know its mathematical 
meaning, yet I'd be all ears :)

> Talking about shortcomings of POV-Ray's incorporation of quaternions there
> is the additional problem that you can't choose the 3d-cut you want to see
> freely ( see the docs; w != 0  !!!), and that the pictures are distorted !
> Last point is easily seen by depicting sqr Julia with c=0; as with complex
> numbers, this defines a 4D-sphere regardless of iteration-depth, and any
> "plane" 3d-cut through a 4D-sphere is a 3D-sphere ( or a point or empty ).
> BUT you get ellipsoids ...

Well, according to the docs, what POV-Ray renders is not the 
3D-hyperplane section itself of the 4D Julia set, but its projection 


hyperplane which had a normal with a null fourth component would be 
orthogonal to the scene's space, and so would end up wholly mapped into 
a single 2D plane, with no volume, and probably devoid of almost all its 
neat geometric intricacies; more or less the same way a 2D picture's 
projection will be squashed into a line if its support plane is 
perpendicular to the image plane.

We can, nevertheless, recover a (rotated in 4D space) copy of every 3D 
section by any hyperplane with a non-zero normal's fourth component 
(even if it should be quite close to zero) by stretching its projection 
along the proper axis; for instance like so:

----------------------------------------------------------------------------

#declare SliceNormal=<2,-3,-1,1>;
#declare SliceOffset=0;

#include "transforms.inc"

julia_fractal{
   <-.2,.2,0,-.3>
   quaternion
   sqr
   max_iteration 200
   precision 100
   slice SliceNormal,SliceOffset
   #declare SliceNormalProj=<SliceNormal.x,SliceNormal.y,SliceNormal.z>;
   #if(vlength(SliceNormalProj)!=0)
     Axial_Scale_Trans(SliceNormalProj, 
sqrt(pow(vlength(SliceNormalProj),2)+pow(SliceNormal.t,2))/SliceNormal.t)
   #end
   pigment{rgb<0,1,0>}
}

----------------------------------------------------------------------------

If you substitute a zero quaternion Julia parameter in this code, you'll 
see the result is a sphere, no matter what SliceNormal you choose, 
provided its fourth component keeps far enough from zero not to 
challenge calculations' accuracy.


> Ah well, I fear that if we say too much, we will be asked to implement that
> ourselves or shut up - that has happened before to very small minorities
> asking for a lot of programmer's work - and rightfully so, I fear.

Well, this thread was by no means intended as a demand! I was just 
curious about the matter. And I'm looking forward to dive into the code 
that controls 4D Julia fractal rendering as soon as I learn C++ :))

> Have a nice weekend

Thanks :)

Regards.

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Frango com Nata <odo### [at] yahoocombr> wrote:
>

> connectivity of Julia sets? Otherwise, I don't know its mathematical
> meaning, yet I'd be all ears :)
>
When using only q(m+1)=q(m)^n+c, that's the ticket. In case of other
formulae, it's a bit more complicated and I just don't have the time right
now ...
suggested reading:
Peitgen&Richter "The Beauty of Fractals", ISBN 0-387-15851-0
Chapter 3, pp 53-55

Concerning your "skewing" of the fractal, I had thought about something like
that myself, but never managed to really do it; other projects came in the
way. I always thought that would be better implemented in the POV source,
so if you get there - have fun !!

best wishes

Karl

Post a reply to this message

"Karl Anders" <kar### [at] webde> wrote:
> When using only q(m+1)=q(m)^n+c, that's the ticket. In case of other
> formulae, it's a bit more complicated and I just don't have the time right
> now ...
> suggested reading:
> Peitgen&Richter "The Beauty of Fractals", ISBN 0-387-15851-0
> Chapter 3, pp 53-55



> Concerning your "skewing" of the fractal, I had thought about something like
> that myself, but never managed to really do it; other projects came in the
> way. I always thought that would be better implemented in the POV source,
> so if you get there - have fun !!


the fractal rotated onto the scene's space, rather than projected upon it,
without having to program it in SDL... Anyway, thanks for your
encouragement :)

Regards.

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