Hi all. I posted it in the "Search for the holy grail" thread, but I think it's worth starting an independent thread. Maybe some of you will start exploring it... It seems to have great variety within it. Potential for crazy fractals similar to the variety of 2d Mandelbrots. At low iterations in the higher n sets (z^n n>6), you find great variety (although it follows the same basic pattern, it just is slightly different, just like 2d mandelbrots).

You could find a z^2 image like Gold Top:

A z^2 like The Fortress of Solitude (this one needs to be rendered at a higher res for full effect):

or the z^5 Diamond Skull (simple one, and lucky on the clipping) :

or the z^13 Wampum Predatorum (clipping not lucky this time, needs re-rendered, recolored, relighted... re-done):

The formula is pretty simple: a rotation of the linear x axis component against the planar y&z component, rotation of the y axis component vs. the x&z planar component, and rotation of the z axis component vs. the x&y planar component.

pixel_x = x component of pixel

pixel_y = y component of pixel

pixel_z = z component of pixel

plane_yz = y&z planar component = sqrt(pixel_y^2 + pixel_z^2)

plane_xz = x&z planar component = sqrt(pixel_x^2 + pixel_z^2)

plane_xy = x&y planar component = sqrt(pixel_x^2 + pixel_y^2)

* If you desire to enter a seed value, you should only set 3 seed values, and use similar equations to the above to create your other 3 seed value components:*

seed_x, seed_y, seed_z <-- self referential

seed_yz = sqrt (seed_y^2 + seed_z^2)

seed_xz = sqrt (seed_x^2 + seed_z^2)

seed_xy = sqrt (seed_x^2 + seed_y^2)

We set x,y, and z to their corresponding components, and the planar variables yz, xz, xy to their corresponding components (for Julias, the first set would be the pixel values, for Mandelbrot type the first set would be seed values).

After our variables are assigned, we do out iteration loop:

Type R2 D2: // All Images in thread are this type, which apparently isn't as "fractal" after further tests

r= sqrt[ (x^2 + y^2 + z^2) ^ n + (yz^2 + xz^2 +xy^2)^n ]

**Type D2:**

r= (x^2+y^2+z^2+yz^2+xz^2+xy^2)^(n/2) *Comparison Shots Follow... Maybe not as varied, but definitely more fractal and symmetric*whiskey= atan2 (x + i*yz) // angle assignment. The names you use for your variables are generally very important.

tango = atan2 (y + i*xz) // or not.

foxtrot = atan2 (z + i*xy)

x= r * cos (whiskey * n) // r is already calculated above

y= r * cos (tango * n)

z= r * cos (foxtrot * n)

yz=r* sin (whiskey * n)

xz=r* sin (tango * n)

xy=r* sin (foxtrot * n)

Then add in either the corresponding pixel or seed components, depending on if you are doing Julia type or Mandelbrot type.

For Mandelbrot: For Julia:

x=x+pixel_x x=x+seed_x

y=y+pixel_y y=y+seed_y

dot dot dot dot dot dot........

xy=xy+plane_xy xy=xy+seed_xy

After all that

rigamorale check your bailout before your next iteration.

I simply bail on |x| + |y| + |z| < 10^30 so I get a smooooooth fractal, although you do need to do a few more iterations for details. Which might explain a thing or 2, perhaps I don't need to do 8 whole iterations to get an insanely detailed 7th order fractal... anyways.

Anyways, there are several other starting magnitudes that I use to generate different fractal types. Some for pyramidal fractals (that I call "Type D3") some for hybrid fractals (fractals that are a cross between d2 and d3), some for Mandala types (which generate really cool patterns at different angular rotations/magnitude settings, but aren't actually fractal), and some for other types, such as this one which I call R2 D2 because it is the 2 radius (magnitude) formula for D2 that produces the most awesome fractals (R2= 2 magnitude).

I'm including this set of z^7 images, because I just found a neat location at the "top" (rotated 45 degrees) of the fractal, which might correspond to the bottom of a quadrant of the fractal... or not. I should check.

The third image below is a zoom into the alter like thing in the center of the second image (and the second is a zoom into basically somewhat below the center of the first). The first is I don't know how many iterations, the second is 9, the third and 4th are 10, and the 5th is the same location as the 4th but 11 iterations.

Calculating one more small iteration increase at the same location before I switch to the search in z^2 (now that I know cool locations are at the "top and bottom" of 45 degree rotated fractal.

Here is the 12 iteration one:

I will upload a few z^2 images later, been exploring z^7 and my z^2s aren't that interesting as I haven't gone that deep in the z^2s in this new magnitude formula (which appears really cool, like all the other ones). I'll also attach a text file that can be used in ChaosPro: it is the set of my formulas, including the one that makes this image "Type R2 D2". I'll get together a help file for it eventually... To use it the FormulaPack in ChaosPro, you just rename it from "Benesi.cfm.txt" to "benesi.cfm" then copy it into the "compiler" directory in the "formula" directory of your chaospro directory (which is in your program files directory, or wherever you installed it).

It has a bunch of my formulas, and I am working on higher dimensional extensions for them that will be included later. Already have a 5d that works, but decided 6d would be better (and haven't totally finished the 5d, so it is not released with these). I decided 6d would be better because I like to set the "time" dimensions to vary with the space dimensions (a little bit, perhaps have arbitrary offset (or starting) values, then vary them by (space dimension)/(some value), one for each space dimension: thus 6 dimensions total).