END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

In case someone is interested, i put together a site, http://sites.google.com/site/fractals3d/home, for what I feel is a more geometrically correct multiplication, operating on geodesics.

--Ingemar Eriksson

Nice, can you post here the formulas?

Yes, I meant the Mandelbrot

Hello all,

I have lurked here for way too long so decided to join the party.

After experimenting with the triplex algebra and alternate formulas for a while now, I have a few things to contribute.

Firstly, maybe this topic should be called "Alternate triplex algebra" as the variations are based on modifying the maths behind the triplex algebra and not the formula z=z^p+c.  After implementing most of Paul's listed variations at the start of this topic I tried a few others;

1. "IQ variation"  This comes from Inigo's page here http://iquilezles.org/www/articles/mandelbulb/mandelbulb.htm.
2. Negative IQ.  Same as IQ but with a negative z compent on the triplex (same as negative SIN compared to SIN)
3. Negative COS.  Same as Daniel's original COS version with a negative z.
4. Negative Rucker.  Again, same as the Rudy Rucker variety with a negative z.

They all give unique results and new bulbs to explore.  There probably are a whole bunch of other variations to explore and render.  If anyone has other ideas for other variations please share.

Some example images of the variations of the power 8 mandelbulb and other formulas here here
http://softology.com.au/gallery/gallerymandelbulb.htm I also have links to my flickr galleries with zoomed in sample images.

The more complex formula varieties do lead to more disconnected and "dusty" fractals (which is logical as it is like comparing the 2D Mandelbrot to the more complex versions of the mandelbrot formula), but when zooming in they do give some unique new looks.

Jason.

The variations I show are all based on the usual triplex algebra at the start of this topic.  The listed algebra is the SIN version;

SIN

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth cosph
sinph

And the other variations have different trig when calculating theta,phi and the x,y,z components of triplex z when doing cartesian to polar conversions, exponents and multiplication.  From what I can see the riemanlettuce uses a different system entirely and creates a more discinnected surface.

-SIN

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth cosph
-sinph

COS

theta=arctan2(z.y,z.x)
phi=arccos(z.z/radius)
costh sinph
sinth sinph
cosph

RUCKER

theta=arctan2(z.y,z.x)
phi=arctan(z.z/z.x)
costh cosph
sinth cosph
sinph

Reversed Order

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth
-costh sinph

Reversed Order Positive Z

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth
costh sinph

IQ

theta=arccos(z.y/radius)
phi=arctan2(z.x,z.z)
sinth sinph
costh
sinth cosph

Negative IQ

theta=arccos(z.y/radius)
phi=arctan2(z.x,z.z)
sinth sinph
costh
sinth cosph

Negative COS

theta=arctan2(z.y,z.x)
phi=arccos(z.z/radius)
costh sinph
sinth sinph
-cosph

Negative Rucker

theta=arctan2(z.y,z.x)
phi=arctan(z.z/z.x)
costh cosph
sinth cosph
-sinph

The variations all seem to work with formulas other than z=z^p+c, so finding new variations leads to a whole bunch of new fractals to explore when using multiple formulas.

I would be interested in any other variations other people have tried using the above style changes.

Jason.

How do I use these variations with the usual triplex algebra?  What part of the calculations do they replace?

To make my previous summaries more complete.  This is for the Nylander Positive SIN version;

(converting cartesian to spherical polar)
radius=sqrt(z.x*z.x+z.y*z.y+z.z*z.z)
theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)

(z^p)
radius=power(radius,p)
theta=theta*p
phi=phi*p
z.x=radius*cos(phi)*cos(theta)
z.y=radius*cos(phi)*sin(theta)
z.z=radius*sin(phi)

Which parts of the calculations do I need to change to implement your variations?

Just tried another idea for getting a "true 3D" Mandelbrot.
Basically I thought for r,i,j then treat r,i and r,j as two complex numbers, raise both to a power, combine the real parts of the results somehow and use the imaginary parts as the new i and j respectively, there was an issue with respect to magnitude but that's simply fixed by scaling the result, here's the UF code I came up with (magn is the magnitude of the triplex on entry):

              magn = magn^@mpwr
              zjk = (real(zri) + flip(zj))^@mpwr
              zri = zri^@mpwr
              zj = imag(zjk)
              x1 = real(zri)*real(zjk)
              if x1>0.0
                zri = x1/sqrt(abs(x1)) + flip(imag(zri))
              else
                zri = flip(imag(zri))
              endif
              x1 = sqrt(|zri| + sqr(zj))
              if x1>0.0
                magn = magn/x1
                zri = magn*zri + cri
                zj = magn*zj + cj
              else
                zri = cri
                zj = cj
              endif
              magn = sqrt(|zri| + sqr(zj))

The result is more similar to the "true 3D" attempts using the unit multiplication table variations than the Mandelbulb so I was going to discard the method completely but I decided to try applying the generic Mandelbox-style square folding and the result was extremely interesting....will post a result in this thread - about 20 minutes to go.

  This is just a variation of the "Christmas Tree Fractal" formula from Paul Nylander's website (that I accidentally recreated).  Anyways, if you don't take the absolute value of the y pixel component, it is more "true" to the classic 2d Mandelbrot, in the sense that the 'distortion' (asymmetry)  introduced by the - x component in the classic 2d Mandebrot for even n (z^2,4,6,8....) is magnified by the y component's interaction with the z component.  I'd think that this was "the real deal", simply because it follows the same pattern of distortion as the 2d Mandelbrot, but it isn't as pleasing as something that replicates the look and feel of a 2d Mandelbrot for even z^n.

Here is the complex triplex version, which should give some insight into the mathematics going on (and it's about 2 times as fast on my CPU):

Trig version of above code, with abs of y pixel component (which introduces symmetry).