Mandelbrot to Mandelbulb - 3D Infinity

So this is the formula then?
What's the formula of this thing?

There are a few subtle variations, which mostly end up producing the same kind of incredible detail. Listed below is one version. Similar to the original 2D Mandelbrot, the 3D formula is defined by:

z -> z^n + c

...but where 'z' and 'c' are hypercomplex ('triplex') numbers, representing Cartesian x, y, and z coordinates. The exponentiation term can be defined by:

[x,y,z]^n = r^n [ sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) ]
...where:
r = sqrt(x^2 + y^2 + z^2)
theta = atan2( sqrt(x^2+y^2), z )
phi = atan2(y,x)

And the addition term in z -> z^n + c is similar to standard complex addition, and is simply defined by:

[x,y,z]+[a,b,c] = [x+a, y+b, z+c]

The rest of the algorithm is similar to the 2D Mandelbrot!

Here is some pseudo code of the above:

r = sqrt(x*x + y*y + z*z )
theta = atan2(sqrt(x*x + y*y) , z)
phi = atan2(y,x)

newx = r^n * sin(theta*n) * cos(phi*n)
newy = r^n * sin(theta*n) * sin(phi*n)
newz = r^n * cos(theta*n)

...where n is the order of the 3D Mandelbulb. Use n=8 to find the exact object in this article.

If infinity is a fractal, then would it actually be expanding by imploding?

heres my favorite 3d mandelbulb watch in 720p

just go here if it doesnt embed properly
www.youtube.com...

[edit on 10-2-2010 by podizzle]

Astonishing!
I just showed this thread to Someone Who I Know, and he told me a lot of that looks like a 'place' he .. 'experienced' whilst under the effect of a certain illegal substance (mentioned by someone else earlier in the thread). Makes you wonder about the 'mind-expanding' side of it.

Anywho, back on-topic...




This image I find particularly interesting.
Stare for a minute - I drew parallels with a sitting Buddha, a Krishna-type thing sat with folded legs - I'm sure theres more, I'll leave it to your imaginations!

S&F for a truly brilliant thread!

reply to post by Evasius


You may think this is strange, but I also believe that the big bang was the conception of a bigger being that we are inside and that the maya calender maybe the countdown from the life to the death of that first Neuron inside it's brain. Anyone think this idea is mad?

reply to post by Evasius

This is absolutely astounding! The Mandelbrot has always been a favorite fractal for me ever since I read the book by James Gleck in 1988.... CHAOS. Some beautiful pictures in the book. But nothing compared to what you have accomplished here. Very beautiful!!!!
This is just amazing. This is by far my favorite thread!!! I would give you infinite flags and stars if I could. Brilliant work!!!

Outstanding thread....watching these 3D paterns give me a difficult to describe feel.

I wonder if there is a connection between nano-tec and fractals possible and what can come out of that.......maybe such a combination can gobble up the universe....?

Anyways, I am very curious about the (useful) applications of these 3D fractals.

Fractals are absolutely amazing. My junior year in high school, we had a early-release day and my Trigonometry teacher had nothing planned so he put on a Nova documentary about Fractals. Everyone else in the class took this as an opportunity to goof off, but I was astounded by them. The minute class got out I went to the library and checked out The Fractal Geometry of Nature by Benoit Mandelbrot and read it. The mathematics involved with fractals are way beyond me, unfortunately, as my math skills are pretty lackluster. However, I do enjoy reading about them and studying them nonetheless.

After visiting this site on numerous occasions I finally decided to make an account.
I wanted to post this as well as tag along in other threads.

www.neave.com...

You can explore it for yourself!

Well if reality is a fractal then Type 0, 1, 2, and 3 civilizations are all contained within us..... they may infact be the foundation for the evolution of all life in the fractal resolutions above.

It all makes sense now. lol, awsome thread though and thank you for this great contribution!

It makes you realize that there is so much more beyond the surface.

[edit on 10-2-2010 by Wisen Heimer]

reply to post by [davinci]

What got me is the Toroidal displays, particularly the top right most.

This is a very good (albeit simplistic) view of the discovered Epi-genome, a secondary DNA strand that encircles our main DNA. Very interesting as research is showing that the epi-genome provides virtually real time genetic change by manipulating gene expression in the main DNA strand (things you experience today will manifest in your grandkids).

The idea that heredity and genetics are guided by fractal geometry is taken very seriously within the scientific community. The fractal building of more and more complexity from relatively simple coding is quite evident. There are numerous scientific papers online and elsewhere exploring the idea, listed below are just a few.

[atsimg]http://files.abovetopsecret.com/images/member/60b0a8f1958c.jpg[/atsimg]

A route to fractal DNA-assembly

Fractal Landscapes and Molecular Evolution: Modeling the Myosin Heavy Chain Gene Family

Fractals related to long DNA sequences and complete genomes

3D fractal DNA assembly from coding, geometry and protection

Compositional segmentation and long-range fractal correlations in DNA sequences

Coding and Geometrical Shapes in Nanostructures: A Fractal DNA-Assembly

Identifying Compositional Domains in DNA Sequences

Thermodynamics of fractal signals based on wavelet analysis: application to fully developed turbulence data and DNA sequences

Fractal properties of DNA walks

Fractal Dimension and Localization of DNA Knots

Fractal landscapes in biological systems

Scaling in Nature: from DNA through Heartbeats to Weather

Fractals and the birth of gothic: reflections on the biologic basis of creativity

Fractal neuronal firing patterns

Here's another collection of scientific animations I've just uploaded (which incorporates quite a few fractals):

Scientific Animation Collection

Believe it or not, fractals manifest themselves in nature.

Fractal patterns spotted in the quantum realm

A scientist from Princeton has evidently just discovered new evidence of fractal geometry in the quantum world - a discovery that was an unintended artifact of his main research.

Ali Yazdani at Princeton University in the US and his colleagues have revealed that these patterns also exist at the scale of individual atoms in a solid. And the key to this effect is a sudden transition where a material changes from a metal to an insulator. At this transition, the waves associated with individual electrons go from being extended across the whole system to being localized at lattice sites.

Talking about his research, Yazdani admits that observing these fractals was not the primary aim of this research. "We do this stuff every day, but once we managed to get the experiment to work with this material, we were confronted with what look like random patterns," he says. His group went on to develop the theory and realized that the electrons they were observing were on the brink of localization.

Image from the article:[atsimg]http://files.abovetopsecret.com/images/member/c00d212982c1.jpg[/atsimg]

[atsimg]http://files.abovetopsecret.com/images/member/a764164c755f.jpg[/atsimg]

larger image

More info:

Electrons on the brink: Fractal patterns may be key to semiconductor magnetism
Princeton News

[edit on 10/2/10 by Evasius]

Originally posted by degoat87
Starred and flagged for sure!

I'm sitting on my University's campus right now with my iPod on shuffle. Interestingly, when I watched the videos of the 3D renders, I felt...a sense of euphoria, regardless of what song came on next. Has anyone else felt this?

Hey I felt the same thing too. It seemed I was in a theta moment with my brain waves etc. Like I was insync with my own consiousness for a brief moment etc

Fractals can also describe economics, especially our current global crisis.

Benoit Mandelbrot and Nassim Taleb off a dire warning



Transcripts of the above interview here:
Top Theorists Examine Rippling Economic Turbulence

World in Chaos and Market Meltdowns, Too Costly To Bear

As we grow older, we realize that life is more like the fractals we studied in last month’s newsletter. Events in the markets and in our lives rarely follow a continuous path of absolute certainty. However, if we step back and look at the big picture, we see… patterns. In the midst of a world that appears to be without order, closer investigation reveals cyclicality. This is the basic thought behind the science and math that has come to be known as Chaos Theory.

“The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale and size, is an attribute of many laws of nature and innumerable phenomena in the world around us. Self-similarity is, in fact, one of the decisive symmetries that shape our universe and our efforts to comprehend it.

Symmetry itself is one of the most fundamental and fruitful concepts of human thought. By symmetry we mean an invariance against change: something stays the same, in spite of some potentially consequential alteration.”

Benoit Mandelbrot and the wildness of financial markets

[atsimg]http://files.abovetopsecret.com/images/member/c18bf35aafd9.jpg[/atsimg]

And as a side-note, a recent article addressing the current Greek 'debt crisis' makes a direct correlation between fractals and the downturn (albeit it in a very non-scientific way):

A Greek crisis is coming to America

Yet the idiosyncrasies of the eurozone should not distract us from the general nature of the fiscal crisis that is now afflicting most western economies. Call it the fractal geometry of debt: the problem is essentially the same from Iceland to Ireland to Britain to the US. It just comes in widely differing sizes.

[edit on 10/2/10 by Evasius]

Here are some obvious fractal representations in nature. More specifically sea nature. One just to have to look at any leaf on a tree to see the obvious signs of fractals are there etc

[edit on 10-2-2010 by Epsillion70]

Aaaaaaaaaamazing thread! Bookmarked to read it all asap!

Thanks for the huge contribution!

reply to post by Evasius


Not only did you blow my mind but you just help me figure out a piece of the puzzle to this thing i'm enjoying called life, thank you. s+f

I am one very happy camper today! I found the best program that utilizes the Mandelbulb. I have a trial version for now, so cannot save any work until I get the registration code but I can do screenshots.

Here is one: If you have 3D glasses (I can't find mine!) this picture will look cooler!
[atsimg]http://files.abovetopsecret.com/images/member/88ae5bacc7e7.jpg[/atsimg]

Program can be found here:
softology.com.au...

and another program that I use a LOT and is free:
www.incendia.net...

I have been looking for something new for weeks! Thank you again for all this information! Why should I settle for dry, bland Mandelbrots when I can have fat and juicy Mandelbulbs!

Peace out from Berkeley and enjoy!