Choosing the squaring formula by location

[kram1032]

It's right above the picture in that same post, and it's kind of valid, but only if you're picky.

I conjecture that the real M3D, if it exists, requires minimal or no symmetrizing.

[KRAFTWERK]

Hello msltoe.

I just wanted to let you know that I managed to get a nice 3D print out of your beautiful formula.
Thank you for the formulas my friends!!! 

I remember so well when I followed this thread and was blown away by that julia, this makes me extra happy now when I can wear it on my finger.



Klick the image for full view! ^

[msltoe]

Johan: Thanks for bringing this formula to life into a beautiful wearable ring.   

[KRAFTWERK]

Johan: Thanks for bringing this formula to life into a beautiful wearable ring.   

Oh, You have no idea how greatful I am for this approval Mike (?).
As I just said to jehovajah: this is the essance of this forum. I just love it, I love you all! 

[M Benesi]

  That does look cool.  I bet a lot of us are thinking about building a CNC machine at some point- want to work with metal, wood, and stone instead of plastics. 

[DarkBeam]

Msltoe one day please share with us the tetrahedral variants

[eiffie]

I am having fun revisiting these as DarkBeam digs them up. I made the formula continuous with some changes but the nice Julias are similar.
WebGL thingy: https://www.shadertoy.com/view/MtfGWM

If the glsl is confusing I could write it out plainly.

[DarkBeam]

Great idea I was looking for the continuous version!

[youhn]

Great stuff! Nice variety of shapes, which are even more beautiful when rotating and morphing. Those WebGL things are great.

I just dove into adding formulas to Mandelbulber v2, so yes please share the formula in the easy form:

newx = ...
newy = ...
newz = ...

This one is an addition to the ones already in MB3D?

MsltoeFoldQuat.m3f
MsltoeSym2.m3f
MsltoeSym3.m3f
MsltoeSym4.m3f
Msltoe_Sym.m3f

Can't check them all, since only Sym2, Sym3 and Sym4 have a description of the formula.

Sym2:

Code:

if abs(y) < abs(z) then exchange(y,z)
if y > z then x = -x
zr = 1 - z*z / (x*x + y*y + z*z)
x' = (x*x - y*y) * zr + Cx
y' = 2*x*y*zr * Y_multiplier + Cy
z' = 2*z * sqrt(x*x + y*y)

Sym3:

Code:

if abs(y) < abs(z) then exchange(y,z)
if y > z then {x = -x, z = -z}  
zr = 1 - z*z / (x*x + y*y + z*z)
x' = (x*x - y*y) * zr + Cx
y' = 2*x*y*zr * Y_multiplier + Cy
z' = 2*z * sqrt(x*x + y*y) + Cz

Sym4:

Code:

if abs(x) < abs(z) * XZ_sym-mul then exchange(x,z)
if abs(x) < abs(y) * XY_sym-mul then exchange(x,y)
if abs(y) < abs(z) * YZ_sym-mul then exchange(y,z)
if (x*z < 0) then z = -z;
if (x*y < 0) then y = -y;
x' = x*x - y*y - z*z + Cx
y' = 2*x*y + Cy
z' = 2*x*z + Cz

[eiffie]

This appears to be different since it is based on rotating yz into -pi/8.0, pi/8.0. What I changed was making this the absolute value so the edges of the "pie slice" align with the next slice. Then I don't rotate it back which may totally ruin the original idea but the julias around y=z=0 look similar.

Code:

float psi = fabs(fmod(atan2(z,y)+pi/8.0,pi/4.0)-pi/8.0);
float r=sqrt(y*y+z*z);
y=cos(psi)*r;
z=sin(psi)*r;
float m=1.0-z*z/(x*x+y*y+z*z);

newx=(x*x-y*y)*m; //also add the julia values here
newy=2.0*x*y*m;
newz=2.0*z*sqrt(x*x+y*y);

[DarkBeam]

Mmm not sure. Cannot help.

Those were early formulas by Jesse.

[cKleinhuis]

awesome dudes, this one is really one that looks like one would expect

but errh, what do you mean by continuous version !?

[DarkBeam]

Chris when a formula is discontinue the distance estimation fails ... you can't derive.
So continuous functions are always preferred

[cKleinhuis]

Chris when a formula is discontinue the distance estimation fails ... you can't derive.
So continuous functions are always preferred

i see, thank you undifferentiable functions are problemetatic, right sir