New Mandelbulb Variations

Rule 52 – Power 8 Mandelbulb

Rule 330 – Power 8 Mandelbulb

Tad Boniecki emailed me with some new Mandelbulb trig variations. The only changes in these from the usual method of rendering Mandelbulbs is the formula for Triplex Power. For the original Positive SIN Mandelbulb the routine looks like;

function SINTriplexPower(const z:triplex;p:double):triplex;
var rcosphi:double;
begin
radius:=power(radius,p);
theta:=theta*p;
phi:=phi*p;
SINTriplexPower.x:=radius*cos(phi)*cos(theta);
SINTriplexPower.y:=radius*cos(phi)*sin(theta);
SINTriplexPower.z:=radius*sin(phi);
end;

The new methods changed the trig values for X, Y and Z. The main change here is there is only a single trig call for each XYZ component.

function Trig1TriplexPower(const z:triplex;p:double):triplex;
begin
radius:=power(radius,p);
theta:=theta*p;
phi:=phi*p;
Trig1TriplexPower.x:=radius*cos(phi);
Trig1TriplexPower.y:=radius*cos(theta);
Trig1TriplexPower.z:=radius*sin(theta);
end;

Theta and Phi are initialised by using

theta:=arctan2(z.y,z.x);
phi:=arcsin(z.z/radius);

After randomly trying different combinations for a while, I wrote a simple loop to go through all possible combinations of these type of Mandelbulb.

For each of the XYZ components they can be in the form +/- sin/cos(phi/theta). eg -cos(phi) or sin(theta) etc. This gives a total of 512 possible combinations.

Using a Wolfram like rule numbering system, convert a nine digit binary number into decimal. The digits of the binary number correspond to;
1. 0 for X COS, 1 for X SIN
2. 0 for X Phi, 1 for X Theta
3. 0 for +, 1 for –
4. 0 for Y COS, 1 for Y SIN
5. 0 for Y Phi, 1 for Y Theta
6. 0 for +, 1 for –
7. 0 for Z COS, 1 for Z SIN
8. 0 for Z Phi, 1 for Z Theta
9. 0 for +, 1 for –
So rule number 20 is binary 000010100 and converts into the formulas
x:=radius*cos(phi);
y:=radius*cos(theta);
z:=radius*sin(phi);

You can see all 512 variations (all power 8 Mandelbulbs) here. Each of the image names show the rule number and corresponding trig calls.

From the 512 possible results, I ignored most of them (too spikey, too similar to others, too lathed, or assymetrical) and picked the 20 most interesting results.

Rule 50