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

Darren New / San Diego, CA, USA (PST)
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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
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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
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On 22Sep08 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?
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Darren New wrote:
> Scroll down to the bottom. The castle fractals are pretty cool.
Personally, I like the glowyplasma stuff... ;)

http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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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 3rdorder
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*
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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 openended 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!
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Orchid XP v8 wrote:
>
> There is another possibility to consider as well: the general 3rdorder
> 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 taffylike
quaternions don't seem like the ultimate destination to me.
Sam
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> 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/
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>> There is another possibility to consider as well: the general
>> 3rdorder 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 POVRay 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 taffylike
> quaternions don't seem like the ultimate destination to me.
Agreed.
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