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) ]
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.
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).
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.
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?
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.”
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.