6D from Fractint! This fractal appeared in number of formula files by Jos Hendriks or Lee H. Skinner, pretty cool shape, kept during zoom on antenna, but alsou zooms are OK, mostly of julia sets, just that iteresting features are rotated what often don't fit in screen and they still are like of mandelbrot set.

Parameter set, zoom on antenna and 3rd dim slice of zHeight=-0.20

Julia sets

(-0.725, 0.135, -0.03, 0)

(-0.9, 0 , -0.17, 0) slice zHeight= (-0.14, 0)

(-0.8, 0, -0.117, 0) slice zHeight= (0.15,0)

(-0.75, 0, 0, 0) this seems to be central in many aspects

(-0.8, 0, 0.117, 0)

Russell Walsmith publicated triternion group 3D multiplication rules and rendered 2D slices of it using Fractint. Like:

z_x= ...

Z_y= ...

Z_z= ...

pdf is still downloadable throught a web archive

https://web.archive.org/web/20040524032840/http://www.fibonacci-arrays.com/Triternions.pdf It is just a rotated by 45 degrees extruded mandelbrot.

Then owning to an elasticity of Fractint's formula language small change to the equation (pixel_y * i) turned each 3D variable into complex equation. Thus came out 6D algebraic group where Z*Z orbit travels throught all the dimensions. I guess, maybe this fractal could lead to some deeper insights about universe, sutch as extra dimension parallel universes or ghostly extraterrestrials of little females.

I multiplied this on paper and made it algebraicaly transparent. Long but easy (repeating):

Initialisation (for this shape Y variable of pixel is put into Yi part):

`Mandelbrot set`

Z_x = 0

Z_xi = 0

Z_y = 0

Z_yi = 0

Z_z = 0

Z_zi = 0

C_x = real(#pixel)

C_yi = imag(#pixel)

C_z = real(@zHeight) ;a variable, slice in 3rd dimension

C_zi = imag(@zHeight) ;a variable, slice in 3rd dimension

Julia set

C_x = real(@julia)

C_yi = imag(@julia)

C_z = real(@zJulia)

C_zi= imag(@zJulia)

Z_x = real(#pixel)

Z_xi = 0

Z_y = 0

Z_yi = imag(#pixel)

Z_z = real(@zHeight) ;a variable, slice in 3rd dimension

Z_zi = imag(@zHeight) ;a variable, slice in 3rd dimension

Iteration:

` tempZ_x = Z_x * Z_x + 2*Z_y * Z_z - Z_xi * Z_xi - 2*Z_yi * Z_zi`

tempZ_y = Z_z * Z_z + 2*Z_x * Z_y - Z_zi * Z_zi - 2*Z_xi * Z_yi

tempZ_z = Z_y * Z_y + 2*Z_z * Z_x - Z_yi * Z_yi - 2*Z_zi * Z_xi

tempZ_xi = 2*Z_x * Z_xi + 2*Z_y * Z_zi + 2*Z_yi * Z_z

tempZ_yi = 2*Z_z * Z_zi + 2*Z_x * Z_yi + 2*Z_xi * Z_y

tempZ_zi = 2*Z_y * Z_yi + 2*Z_z * Z_xi + 2*Z_zi * Z_x

Z_x = tempZ_x + C_x

Z_xi = tempZ_xi + C_xi ;here c=0

Z_y = tempZ_y + C_y ;here c=0

Z_yi = tempZ_yi + C_yi

Z_z = tempZ_z + C_z

Z_zi = tempZ_zi + C_zi

;make complex Z (for color methods):

Z = Z_x + flip(Z_y)

bailout:

`abs(Z_x) + abs(Z_xi) + abs(Z_x) + abs(Z_y) + abs(Z_yi) + abs(Z_z) + abs(Z_zi) < @bailout`

There were other versions, less smooth. Here is the original Fractint formula, each variable is automaticaly turned complex by addition of the "*(0,1)":

somename { ; Jos Hendriks <***@hexaedre-fr.com>

; Fri, 24 Jan 2003 21:38:21

; Lee H. Skinner <***@thuntek.net>

; Wed, 22 Jan 2003 10:27:11

;

c1=real(pixel),c2=imag(pixel)*(0,1),c3=p1

z1=z2=z3=0:

t1=z1*z1+2*z2*z3

t2=z3*z3+2*z1*z2

t3=z2*z2+2*z3*z1

z1=t1+c1,z2=t2+c2,z3=t3+c3

z=(z1^2+z2^2+z3^2)^.5

z < 16

}

It is somewhat turned by 45 degrees, so without rotation often don't go well in screen but the result is interesting. Alsou it turns by 90 degrees, is have some connection to split-complex aka minkowski space - square mandelbrot set. Set must be disconected in 2D becouse it is connected in higher number of dimensions. It appears, disapears and then repeats.

Shapes into mandelbrot set shapes:

I would like to see what it is in 3D but so far I was too lazy to implement that. Maybe later if I have an inspiration.

p.s.

"Did the Mandelbrot set exist before computers were invented?"