POV-Ray : Newsgroups : povray.off-topic : Cool 3D fractals Server Time
27 Oct 2021 06:08:19 EDT (-0400)
  Cool 3D fractals (Message 1 to 10 of 10)  
From: Darren New
Subject: Cool 3D fractals
Date: 22 Sep 2008 13:14:56
Message: <48d7d290$1@news.povray.org>
http://www.skytopia.com/project/fractal/mandelbrot.html

Scroll down to the bottom. The castle fractals are pretty cool.

-- 
Darren New / San Diego, CA, USA (PST)


Post a reply to this message

From: Doctor John
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 13:39:11
Message: <48d7d83f$1@news.povray.org>
Darren New wrote:
> http://www.skytopia.com/project/fractal/mandelbrot.html
> 
> Scroll down to the bottom. The castle fractals are pretty cool.
> 
A little bit of an understatement, methinks

John

-- 
"Eppur si muove" - Galileo Galilei


Post a reply to this message

From: stbenge
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 15:23:25
Message: <48d7f0ad@news.povray.org>
Doctor John wrote:
> Darren New wrote:
>> http://www.skytopia.com/project/fractal/mandelbrot.html
>>
>> Scroll down to the bottom. The castle fractals are pretty cool.
>>
> A little bit of an understatement, methinks
> 
> John

I came across this page just yesterday! I wonder if anyone here thinks 
there is really a 3D Mandelbrot as the author describes?

And those castle fractals were pretty cool. I thought they were POV at 
first, but are in fact Xeno Dream.

Sam


Post a reply to this message

From: andrel
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 16:16:02
Message: <48D7FD4C.8020505@hotmail.com>
On 22-Sep-08 19:14, Darren New wrote:
> http://www.skytopia.com/project/fractal/mandelbrot.html
> 
> Scroll down to the bottom. The castle fractals are pretty cool.
> 
do paul bourke and rune (and perhaps others here) know that are on that 
page?


Post a reply to this message

From: Orchid XP v8
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 16:34:05
Message: <48d8013d$1@news.povray.org>
Darren New wrote:

> Scroll down to the bottom. The castle fractals are pretty cool.

Personally, I like the glowy-plasma stuff... ;-)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*


Post a reply to this message

From: Orchid XP v8
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 16:45:09
Message: <48d803d5$1@news.povray.org>
stbenge wrote:

> I wonder if anyone here thinks 
> there is really a 3D Mandelbrot as the author describes?

It's a basic fact that you can't generalise complex algebra to more 
dimensions and still have a "field"; one or other of the axioms must be 
broken. The result is the hypercomplex and quaternion algebras, which 
just look like (optionally twisted) surfaces of revolution.

So you can't do anything to the generating formula that produces a 
nontrivially 3D image. But the 2D set has many obvious geometric 
properties (particularly the prominent appearence of circles). Could you 
not manually reproduce those same relationships with spheres instead? I 
think perhaps you could. Working out what to do with all the "filaments" 
would be harder, but not in principle impossible.

There is another possibility to consider as well: the general 3rd-order 
complex equation has *two* unknowns instead of one, resulting in 2D 
Julia sets but a 4D Mandelbrot set. So here we have a 4D set based on a 
true field algebra, which structure in all directions. And it follows 
the same kind of patterns as the 2nd order set. Maybe this could be 
interesting to explore?



A more important question: If this mythical set actually exists, would 
it be interesting to look at?

Draw a tangled mess of lines on a sheet of paper and the human brain is 
very good at untangling it. But draw a tangle of lines in 3D and 
suddenly it just looks like a mess.

I rather suspect that any 3D object with an intricate fractal structure 
to its surface is likely to just look random and chaotic and rather 
uninteresting. For example, go pick up a sponge and look at it. Does it 
look interesting? Not really. It just looks like a uniform fuzzy mass. 
Similarly for a lump of bread.

For a 3D fractal to *look* good, its surface would have to be 
sufficiently "simple" that the brain can comprehend it. The brain 
doesn't seem to respond to surface textures as precisely as it responds 
to intricate colours.



Just my thoughts on the matter...

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*


Post a reply to this message

From: Kevin Wampler
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 17:11:01
Message: <48d809e5@news.povray.org>
Orchid XP v8 wrote:

> So you can't do anything to the generating formula that produces a 
> nontrivially 3D image.

I don't see why not, the generating formula only involves squaring and 
addition (no division) so it don't necessarily matter if you're doing 
the computation over a field or not.  That said, it's possible that some 
of the geometric properties of the set arise for reasons intricately 
linked to the field nature of the complex numbers, but I really don't 
know if this is actually the case or not.

> A more important question: If this mythical set actually exists, would 
> it be interesting to look at?

Well, the author of the webpage had "interesting to look at" as more or 
less the definition of what he was looking for, so I think from his 
perspective the question is more like "is there some number system which 
produces an interesting 3D version of the Mandelbrot set".  There should 
be many ways to define multiplication and addition over 3+D points if 
all you care about is the end result and not what axioms they satisfy, 
so I think it's a pretty open-ended question.

> I rather suspect that any 3D object with an intricate fractal structure 
> to its surface is likely to just look random and chaotic and rather 
> uninteresting. For example, go pick up a sponge and look at it. Does it 
> look interesting? Not really. It just looks like a uniform fuzzy mass. 
> Similarly for a lump of bread.
> 
> For a 3D fractal to *look* good, its surface would have to be 
> sufficiently "simple" that the brain can comprehend it. The brain 
> doesn't seem to respond to surface textures as precisely as it responds 
> to intricate colours.

He gave some interesting 3D pictures of things which had the sort of 
structures he was looking for, so in principle I think it's possible.  I 
do, however, tend to agree with you that it's much harder to get 
something fractally looking good in 3D than in 2D or 2.5D, but in some 
ways that's what makes it an interesting challenge!


Post a reply to this message

From: stbenge
Subject: Re: Cool 3D fractals
Date: 22 Sep 2008 19:03:37
Message: <48d82449@news.povray.org>
Orchid XP v8 wrote:
> 
> There is another possibility to consider as well: the general 3rd-order 
> complex equation has *two* unknowns instead of one, resulting in 2D 
> Julia sets but a 4D Mandelbrot set. So here we have a 4D set based on a 
> true field algebra, which structure in all directions. And it follows 
> the same kind of patterns as the 2nd order set. Maybe this could be 
> interesting to explore?

Are you talking about extruding a 2D fractal along a third axis and 
varying the values? Is this not what ends up producing those bubble gum 
shapes? It sounds like it, but then again, my math skills do not allow 
me to visualize what you are saying. Is this a new concept? If so, you 
might find your name in a fractal news journal somewhere if you apply 
the concept :)

> A more important question: If this mythical set actually exists, would 
> it be interesting to look at?
> 
> Draw a tangled mess of lines on a sheet of paper and the human brain is 
> very good at untangling it. But draw a tangle of lines in 3D and 
> suddenly it just looks like a mess.
> 
> I rather suspect that any 3D object with an intricate fractal structure 
> to its surface is likely to just look random and chaotic and rather 
> uninteresting. For example, go pick up a sponge and look at it. Does it 
> look interesting? Not really. It just looks like a uniform fuzzy mass. 
> Similarly for a lump of bread.
> 
> For a 3D fractal to *look* good, its surface would have to be 
> sufficiently "simple" that the brain can comprehend it. The brain 
> doesn't seem to respond to surface textures as precisely as it responds 
> to intricate colours.

Perhaps shading is the key. The 2D Mandelbrot makes sense to the eye 
primarily because of the black basin, and then the colors depicting 
iterations after that. So how would this work in three dimensions? The 
basin might extend from itself with branching structures, with certain 
areas of prominence. The whole thing would look confusing unless you 
applied a light shining down upon it. Or maybe each iteration could be 
made translucent, which might work fairly well, though you would have to 
increase the transparency if you wanted to zoom in further. Of course 
all this is moot unless you figure out out how to apply true 
transcendental complexity to the third dimension. Those taffy-like 
quaternions don't seem like the ultimate destination to me.

Sam


Post a reply to this message

From: scott
Subject: Re: Cool 3D fractals
Date: 23 Sep 2008 03:32:09
Message: <48d89b79$1@news.povray.org>
> Draw a tangled mess of lines on a sheet of paper and the human brain is 
> very good at untangling it. 

You mean like completing level 20 on Planarity? ;-)

http://www.planarity.net/


Post a reply to this message

From: Invisible
Subject: Re: Cool 3D fractals
Date: 23 Sep 2008 04:05:35
Message: <48d8a34f@news.povray.org>
>> There is another possibility to consider as well: the general 
>> 3rd-order complex equation has *two* unknowns instead of one, 
>> resulting in 2D Julia sets but a 4D Mandelbrot set. So here we have a 
>> 4D set based on a true field algebra, which structure in all 
>> directions. And it follows the same kind of patterns as the 2nd order 
>> set. Maybe this could be interesting to explore?
> 
> Are you talking about extruding a 2D fractal along a third axis and 
> varying the values? Is this not what ends up producing those bubble gum 
> shapes?

No. Those are produced by iterating Z = Z^2 + C, but with Z and C as 
hypercomplex or quaternion numbers instead of the usual complex numbers.

What *I* am talking about is iterating Z = Z^3 - 3 A^2 Z + B, where Z, A 
and B are all normal complex numbers. A Julia set is rendered by varying 
the start value for Z - which still has 2 components (Re(Z) and Im(Z)). 
However, the Mandelbrot set is drawn by varying the parameters, which 
gives us 4 axies: Re(A), Im(A), Re(B) and Im(B).

This is not new, just not very widely known. A few people have drawn it 
before. If you search *waaay* back through the POV-Ray images newsgroup 
you'll find some renderings I did.

As I say, it turns out that the higher the iteration count (and hence 
the more complex the surface), the less "interesting" the image actually 
becomes.

> Perhaps shading is the key. The 2D Mandelbrot makes sense to the eye 
> primarily because of the black basin, and then the colors depicting 
> iterations after that. So how would this work in three dimensions? The 
> basin might extend from itself with branching structures, with certain 
> areas of prominence. The whole thing would look confusing unless you 
> applied a light shining down upon it. Or maybe each iteration could be 
> made translucent, which might work fairly well, though you would have to 
> increase the transparency if you wanted to zoom in further.

All of that seems at least plausible.

> Of course 
> all this is moot unless you figure out out how to apply true 
> transcendental complexity to the third dimension. Those taffy-like 
> quaternions don't seem like the ultimate destination to me.

Agreed.


Post a reply to this message

Copyright 2003-2021 Persistence of Vision Raytracer Pty. Ltd.