Another shot at the holy grail

[quick-dudley]

I forget what inspired me to do away with i, but here's the basic result:



It uses two pseudo-imaginary numbers which follow this multiplication table:

	1	m	n
1	1	m	n
m	m	n	-1
n	n	-1	m

For:
For a number space that doesn't use i, it's a surprisingly natural three dimensional generalisation of complex numbers
The cross-section y=z is the mandelbrot set (although slanted)
Some areas are complex and compelling even outside the cross-section y=z

Against:
Some surfaces are too smooth.
Some areas look like candyfloss
It is not symmetrical.

Here is another view...

[DarkBeam]

Is someone able to convert it to a squaring rule?
Like...
Nx= x^2-y^2+cx
Ny=xy+cy
...
Luca

[lkmitch]

It uses two pseudo-imaginary numbers which follow this multiplication table:

   1   m   n
1   1   m   n
m   m   n   -1
n   n   -1   m

If T = x + ym + zn, then
T^2 = (x + ym + zn)*(x + ym + zn)
= x^2 + xym + xzn + xym + y^2m^2 + yzmn + xzn + yzmn + z^2n^2.
Using the multiplication table,
T^2 = x^2 - 2yz + m(2xy + z^2) + n(-2xz + y^2)
real part: x^2 - 2yz
m part: 2xy + z^2
n part: -2xz + y^2

[DarkBeam]

Thanks dear sir 

[youhn]

Is this correct?

Code:

double x2 = z.x * z.x;
double y2 = z.y * z.y;
double z2 = z.z * z.z;
double newx = x2 - 2 * z.y * z.z;
double newy = z2 + 2 * z.x * z.y;
double newz = y2 - 2 * z.x * z.z;
z.x = newx;
z.y = newy;
z.z = newz;

Some images of rotating around it:



Created with hacked version of Mandelbulber v2.02.

[youhn]

After comparing the images, it does NOT seem to be the same thing. Is it a mistake to map the Re, m, n parts to X, Y, Z axis in Mandelbulber?

{additional edit}

It has celtic cross like structures:

[eiffie]

From the multiplication table I get newz=2*x*z+y^2 but I like the celtic crosses so I left it as -2*x*z in this webgl animation thingy...
https://www.shadertoy.com/view/MtsGD4

[DarkBeam]

Just noticed this fmla is 3 years old!
Then go digging again!

[DarkBeam]

I wonder if it's raelly worth it to implement since there's already a "Makin3D-2" formula in MB3D almost identical;

x' = +x*x + z*y*2 + Cx
y' = -y*y - z*x*2 + Cy
z' = -z*z + x*y*2 + Cz

[DarkBeam]

Yeah I implemended Kerry Mitchell's formula and the correct one. Awesome teamwork thanks

Param showing a crossy Julia;

Mandelbulb3Dv18{
g.....S....O/...w....2....kpdAwlMZd0.9IgT8dZ/f1EuOFyVBMV.wP2eBq7ubRqzU2przhyeExj
................................psPTcndFpz1........A./..................y.2...wD
...Uz6.....0..../M.0/.....EN/...U/....E3.....AIEqGuCZhqD/..........m/dkpXm1....U
z.EnAnYD12../2Exo8FPArVzzmgFFK4dHmpDeQH9kORsax9...........U0.....y1...sD...../..
.zXaNadDs2TOWiHNfx1..........SSDKsMDlNdjSQ4Rz.cM0r9ClbacvIqOzwwGokMDlNdjFj2BAqHQ
KunuxMVXx2bZy.HcR.uCZhqDU.....Y5..............sD.6....sD..G.....................
.............oAnAt1...sD....z...........................................C....k1.
.....Ksulz1.......kz.wzzz1.U..6.P....U5...EB....m....c3....F....6/...I1.....SF52
...U.qFG9yzb2zzzRYoWzz7lz16.pc..zXCc..kvrEtMc7xD6ocyFE0ujz1..........2.28.kFrA0.
.Ub96aAIVz9.1se7Umvxz0........../EU0.wzzz1...........s/...................E.2c..
zzzz.............0...................2./8.kzzzD............8....................
/EU0.wzzz1...................................2CcN/UvNPcveeWCNq0.yRiibHJJUk1f..XR
SvBmx3CcN/UvNPcvQsLsUa3.ibhVi1bTV1OK.sSq4uCly3CcN/UvNPcvMwLsUa3.ibhVinqTV1OK.sSq
4uCkz3CcN/UvNPcv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................
E....2..F2E.....I.........EIpZqMfFIRYlKNthIH.........................c..........
...................6./........zj................................................
................................................................................
........................}
{Titel: Crosses in KM~}

In the "correct brot" you may find nice spots near the minibrots

Mandelbulb3Dv18{
g.....m....V1...w....2....UYsOp7DW1ozWzGNzwA5Z0EsCl1eu2ivzfbvek7PA5VzSiFJFK5MUuj
................................OFjP0RNKr.2........A./..................y.2...wD
...Uz6........../M.0/.....UE/...r/....E3.....UawVZ0IorlD/.......nee70dkpXm1....U
z.UaNaVD12../.Exo8FPArVzzmgFFK4dHmpDeQH9kORsax9...........U0.....y1...sD...../..
.zXaNadDr.jvP05lsvHOD6zrh6Y3z.wdVGcgEAljqN4wrKQUvv9ko/55Zi.3z.03KBiOfIlD26Z1llHT
QwnikXxIJHZ9yEg58gQo/8kDU.....o6..............sD.6....sD..G.....................
.............oAnAt1...sD....z...........................................O....k1.
.....Ksulz1.......kz.wzzz1.U..6.P....U5...EB....m....c3....F....6/...I1.....SF52
...U.qFG9yzb2zzzRYoWzz7lz16.pc..zXCc..kvrEtMc7xD6ocyFE0ujz1..........2.28.kFrA0.
.Ub96aAIVz9.1se7Umvxz0........../EU0.wzzz1...........s/...................E.2c..
zzzz.............0...................2./8.kzzzD............8....................
/EU0.wzzz1...................................2CcN/UvNPcveeWCNq0.yRiibHJJUk1f..XR
SvBmx3CcN/UvNPcvQsLsUa3.ibhVi1bTV1OK.sSq4uCly3CcN/UvNPcvMwLsUa3.ibhVinqTV1OK.sSq
4uCkz3CcN/UvNPcv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................
E....2..F2E.....I.........EIpZqMfFIRYlKNt/EH.........................c..........
...................6./........zj................................................
................................................................................
........................}
{Titel: Minibrots~}

Cool spirals (in 3D) in a Julia:

Mandelbulb3Dv18{
g.....m....V1...w....2....k.dNtrZgc/.940MsEjO41EAT2qgukKHz99VgLDbUjYzsWtG/C.Gmwj
................................oAeYyethxz1........A./..................y.2...wD
...Uz6.....A..../M.0/.....ET0...u/....E3.....wngt3gG7SpD/.......nee70dkpXm1....U
z.UaNaVD12../2...........cNaNaNaNavj........Uz9...........U0.....y1...sD...../..
.zXaNadDrrh7esSRnw1/xxtx5pLIzYd9yc3uprojDzcjNuDbpwPuyWO82/sHz.OaieoDW0pDfI4IHFdM
Kx1y0WLhVIxNysSNWQBMJonDU.....I./.............sD.6....sD..G.....................
.............oAnAt1...sD....z...........................................M....k1.
.....Ksulz1.......kz.wzzz1.U..6.P....U5...EB....m....c3....F....6/...I1.....SF52
...U.qFG9yzb2zzzRYoWzz7lz16.pc..zXCc..kvrEtMc7xD6ocyFE0ujz1..........2.28.kFrA0.
.Ub96aAIVz9.1se7Umvxz0........../EU0.wzzz1...........s/...................E.2c..
zzzz.............0...................2./8.kzzzD............8....................
/EU0.wzzz1...................................2CcN/UvNPcveeWCNq0.yRiibHJJUk1f..XR
SvBmx3CcN/UvNPcvQsLsUa3.ibhVi1bTV1OK.sSq4uCly3CcN/UvNPcvMwLsUa3.ibhVinqTV1OK.sSq
4uCkz3CcN/UvNPcv..EsUa3feeWCNqGQIJ36wk8EwyLsUa3f................................
E....2..F2E.....I.........EIpZqMfFIRYlKNt/EH.........................c..........
...................6./........zj................................................
................................................................................
........................}
{Titel: Spirals}

[youhn]

Here's a picture from the correct formula from quick-dudley :



Created with slightly altered version of Mandelbulber v2.02.

[youhn]

Another one from the quick-dudley version. A 3D julia.



Source: http://jeroensnake.deviantart.com/art/Filament-cleaning-ship-510226673

[matsoljare]

Whoa, let's see that one rotated....

[youhn]

Rotated or rotating? The first I actually rendered before the one above, but I didn't like the colors. If you mean rotating as in animated ... don't count on me. My expectations of image quality do not (yet) match with my patience and computational speed of computers.



Not sure if this is the exact same location. Rotation 45 deg, roll of 90 deg.

[Alef]

Cool visuals but no. Too stretched.